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This paper continues the study of the smoothness properties of (real) topological linear spaces. First, the smoothness results previously obtained about various important classes of locally convex spaces, such as Schwartz spaces, are improved. Then, following the ideas of Bonic and Frampton, we use these results to give sufficient conditions for the existence of smooth partitions of unity on manifolds modelled on topological linear spaces. 1. Preliminaries. In order to make this paper self-contained we include the definition of the two types of derivative used and the definition of an 8-category. We will employ the definitions of the derivative in topological linear spaces investigated in detail by Averbukh and Smolyanov [2], [31. See also [14], [15] and [191. Let TLS denote the class of all Hausdorff topological linear spaces over the real field R. Let 2 1(E, F) = 2(E, F) denote the set of all continuous linear maps from E into F, where E, F e TLS. We define by induction p(E, F) = 2(E, p 1(E, F)). Each ?1p(E, F) is given the topology of uniform convergence on bounded subsets of E. If X is a topological space, W(X) will denote the class of all open subsets of X. Let f: U -' V, where U e ((E), V E C(F), E e TLS and F c TLS. Then we say f is Fre6cbet differentiable at x e U, if there exists u e S(E, F) such that for each bounded subset B of E and for each o-neighbourhood W in F, there exists 8 > 0 such that f(x + th) f(x) u * th E tW, whenever b E B and ItL

In this paper we develop algebraic and geometric conditions which imply that a given proper Dehn map can be replaced by an embedding. The embedding, whose existence is implied by our theorem, retains most of the algebraic and geometric properties required in the original proper Dehn map. I. Preliminaries. In this paper all spaces will be simplicial complexes and all maps will be piecewise linear. The boundary, closure, and interior of a subset X of a space Y will be denoted by bd (X), cl (X) and int (X), respectively. A proper map f, taking a space X into a space Y, is a map such that f bd (X) = bd (Y) n f(X). A Dehn map f, taking a compact, bounded, two-manifold F into a 3-manifold M, is a map such that (1) bd (F)=f-lfbd (F); (2) fIbd (F) is a homeomorphism. All unlabeled homomorphisms are natural maps induced by inclusion. We shall consider the problem of replacing a proper Dehn map of a compact, connected, orientable, bounded surface F into a compact, (necessarily) bounded, irreducible, orientable 3-manifold M with an embedding. As will be pointed out later, the requirement of orientability may not be strictly necessary. The author would like to thank J. Gross for a number of helpful conversations. In [3] Papakyriakopoulos proved Dehn's lemma, that is, a proper Dehn map f of a disk 9 into a 3-manifold M can be replaced by a proper embedding g of 9 into M such that g bd (9) =f bd (9). A natural question to consider is the following: Let f: F -* M be a proper Dehn map of a compact, connected, bounded, orientable surface F into a compact 3-manifold M. Does there always exist a proper embedding g: F1 -? M such that (1) g(bd (F1))'f(bd (F)); (2) genus (F1)

Introduction. The concept of isotopy, recently introduced(') by A. A. Albert in connection with the theory of linear non-associative algebras, appears to have its value in the theory of quasigroups. Conversely, the author has been able to use quasigroups(2) in the study of linear non-associative algebras. The present paper is primarily intended as an illustration of the usefulness of isotopy in quasigroup-theory and as groundwork for a later paper on algebras, but is bounded by neither of these aspects. The first two sections are devoted to the basic definitions of quasigroup and isotopy, along with some elementary remarks and two fundamental theorems due to Albert. Then there is initiated a study of special types of quasigroup, beginning with quasigroups with the inverse property (I. P. quasigroups). A system Q of elements a, b, * * * is called an I. P. quasigroup if it possesses a single-valued binary operation ab and there exist two one-to-one reversible mappings L and R of Q on itself such that

Introduction. Let D be an arbitrary domain on the Riemann sphere (that is, D is an open connected set on the sphere) and let p be a point of D. Let H '(D) denote the space of bounded analytic functions on D. We consider the problem of maximizing If'(p)l for fin H-(D) with If(z)lj< 1 for all z in D, and f(p) =0. In the case when D is the unit disc U and p =0, this is Schwarz's lemma and the solution is classical: the maximum is one and if If'(0) = 1, then f(z) = eiOz for some real constant 0. In the general case, the theory of normal families assures us that there is at least one fo in H (D) with foll? 1 I and fo(p) ? f'(p)I for all f in H (D) bounded by 1 which vanish at p. We call any such function extremal. Theorem 1 of this paper provides an elementary proof that there is only one extremal function. We thus add to our problem the question of investigating the properties of the extremal function. In the case when D is bounded by a finite number of disjoint analytic simple closed curves, the properties of the extremal function have been studied extensively (see [1] and [7]). In this case, the extremal function is known to be analytic over the boundary and to have modulus one there. We show that this is a local property and carries over to all points in the boundary of D where the boundary is sufficiently well behaved, no matter what the remainder of the boundary of D is like. In ?1 we also show that the extremal function is closely related to the problem of removable singularities for bounded analytic functions. In ?2 we discuss a related extremal problem and show that the properties of its solution imply that on many planar domains D, each function analytic and bounded by 1 on D may be approximated uniformly on compact subsets of D by a sequence of inner functions. We also briefly discuss extensions of the results to open Riemann surfaces. Finally, some remarks on previous work on Schwarz's lemma are in order. Carleson [3, pp. 78-82] and Havinson [8, Theorem 9] have previously shown that the extremal function is unique. Our proof differs substantially from each of theirs and is considerably more elementary. Furthermore, it does not make use of any of the results from the finitely-connected case. Also, our Theorems 2 and 3 and a weaker version of Theorem 4 have previously been proved by Havinson [8, Theorems 22, 28, and 20]; however, as in the case of Theorem 1, the proofs given here are appreciably simpler and more elementary. Indeed, part of the motivation for this paper was to find simpler and more transparent proofs of these results. In particular, it is only in the proof of Theorem 5, which describes the behavior

Introduction. In a recent paper [17] we obtained a spectral representation for the bounded normal operator solutions of a certain functional equation whose special cases include the semigroup equation (U,,, = U,LUV) as well as many others. In this note we continue our study of the abstract functional equation expressed in terms of algebras of measures. We shall consider therefore a mapping z -* UZ of a locally compact space Z into the set of unbounded normal operators on a Hilbert space. Since the operators are not necessarily bounded, the conditions imposed on the mapping z -* Uz vary somewhat from the conditions in the bounded case. The main result (Theorem 1) which we obtain here is a spectral representation analogous to the result valid in the bounded case [17; 18, Theorem 2, p. 8]. The methods which we use are suggested by the papers [12] and [13] of C. Ionescu Tulcea. For the general theory and terminology of unbounded operators we refer the reader to the works of E. Hille and R. Phillips [10], C. Ionescu Tulcea [11], F. Riesz and B. Sz.-Nagy [21]. Spectral representations for semigroups of unbounded operators are also discussed in A. Devinatz [3], A. Devinatz and A. E. Nussbaum [4; 5], R. Getoor [7], C. Ionescu Tulcea and A. Simon [14] and A. E. Nussbaum [19]. Theorem 1 of this paper contains the main part of Theorem 1 in [14] (and henceforth certain of the results given in [3; 4; 5; 7; 19]). The setting of the problem in algebras of measures is given in ??1, 2, and 5. Spectral families of Radon measures and direct sums thereof are summarized in ??3 and 4. The main result is stated and proved in ?6. In ?8 we apply the main theorem to derive a corollary which in the bounded case extends a result of S. Kurepa [16].

Introduction. In the preceding paper [8] we proved that if all harmonic functions in n-dimensional bounded domain D satisfy the mean value property (MVP) with respect to some point P and a given density function ,t (volumedensity, surface-density, etc.) then, under some simple assumptions on ,, D must necessarily be a ball with center P. In the present paper we are interested in studying the MVP from a complementary point of view, namely, we are interested in finding conditions on ,u (u>?0) under which the MVP holds at most for a finite number of linearly independent functions. The MVP is meant to be

Introduction. Problems of great interest in real-variable theory are the analysis of the structure of a function whose derivative is finite, and the determination of a function when its derivative is given and is finite at each point. The study of these problems has led to a theory of non-absolutely convergent integrals.t The corresponding problems when the derivative in question is with respect to a function of bounded variation have received little attention, and it is with these that the present paper is mainly concerned. In this case we find that there is involved a theory of non-absolutely convergent integrals with respect to a function of bounded variation. Lebesguet has given these questions some consideration. His results depend, in part, on a transformation which changes integration with respect to a function into ordinary integration. By means of this transformation very general results are easily obtained. But for dealing with some particular situations it is not altogether suitable; for example, a discussion of integral equations which involve integration with respect to a function of bounded variation. For this reason we have, in the present work, adhered to methods which are direct. Incidental to our main purpose is a study of the derivative of a function F(x) with respect to a function of bounded variation a(x). Here, too, we have been preceded by others, chiefly Daniell.? In the Transactions paper Daniell concerns himself with the central derivative with respect to a

Suppose C is a compact subset of the plane having a piecewise smooth boundary AC. Let F(r, 0) be the Fourier transform, in polar coordinates, of the indicator function of the set C, where by the indicator function of C, we mean the function whose value on C is 1, and whose value on the complement of C is 0. In ?1 of this paper, we shall describe some relationships between geometric properties of C, and the asymptotic behavior of F(r, 0) as r -x- 00. In ?2, we shall give applications of the results of ?1 to some questions in the geometry of numbers. 1. If AC is sufficiently smooth, and has everywhere positive Gaussian curvature, it is known that the function 'D(6) = sup, r312IF(r, 0)1 is bounded on S' (cf. [1]). If AC has points of zero curvature, this need no longer be true (cf. [3]). The following, however, remains true: THEOREM 1. If AC is of class Cn + 3, for some integer n 1, and if the Gaussian curvature of AC is nonzero at all points of AC, with the possible exception of a finite set, at each point of which the tangent line has contact of order 1. Moreover, 'D(6) is always bounded, except in neighborhoods of those points of S' which, regarded as vectors, correspond to exterior or interior normals to AC at points of zero curvature. In a neighborhood of such a point 006 'D(6) is bounded by a multiple of [dist (6, 00)] -n - 1)/2n , where dist (6, 00) is the length of the smaller arc of S' connecting 0 and 6o, and nj is the largest order of contact which can occur between AC and its tangent line, at those points of AC at which the exterior normal is either 00 or- o REMARK. Theorem 1 has analogues in higher dimensions. I shall show in another paper, by different methods, that if C is a compact convex subset of Rn, whose boundary is analytic, and if F(r, 0) is the Fourier transform, in polar coordinates, of the indicator function of the set C, then supr r(n + 1)/21 F(r, 0)1 is of class LP on Sn1, for some p>2. If C is a polygon, the estimates are of a quite different character. THEOREM 2. Suppose C is a polygon. Then

The importance and usefulness of Taylor's theorem need not be dwelt upon here. We are interested in it for functions whose arguments and functional values belong to abstract spaces of the Frechet type. Consequently no Rolle's theorem can be even stated, and the proofs will be somewhat different from the usual proofs in the theory of numerically-valued functions. It is also to be expected that a slight strengthening of hypotheses will be required. However, it is not necessary to assume the existence of a uniformly continuous nth differential, as was done in the first announcement of these results.: It is sufficient that the function should have an nth variation (in the sense of Gateaux), with certain limitations on its discontinuities. The functions we discuss will be one-valued functions whose arguments x and functional values y belong to linear metric spaces X and ) respectively, briefly, functions F on X to S9. Linear metric spaces correspond to the spaces called by Frechet, "espaces (D) vectoriels." For the notations, postulates and fundamental propositions, we refer the reader to Part 1 of a paper by T. H. Hildebrandt and the author,? to avoid practically entire repetition of that section. No other parts of that paper are needed here. We shall consistently use the letters r, s, a, b, c to refer to real numbers, and the German t to refer to the real axis. The notation To will then refer to a set of open intervals of that axis. By the notation (ab) we shall mean the bounded closed interval of St with end points a and b. Whenever the space M) is required to be complete, this will be specifically mentioned. The form of remainder obtained in our Taylor's formula is a generalization of that given by Jordan|| and analogous at least to those obtained in

Introduction. Let G be a locally compact topological group which is unimodular, that is, the left and right Haar measures coincide. We propose, in this and subsequent papers, to investigate the structure of its "L2-system." By the "L2-system" we mean the (complex) Hilbert space formed from the complex-valued measurable functions f on G for which If| 2 is integrable with respect to the Haar measure, and with a product defined in terms of convolution. This product is not always defined (for that reason we call it an L2-system rather than an L2-algebra) but there are dense subsets (for example, L,nL2) on which multiplication always is defined so it is "almost" an algebra. Our theorems depend only on certain key properties of these L2-systems so we shall investigate general systems with these properties, calling them H-systems. We do this not only for generality but because when decomposing an L2-system into minimal parts we expect those parts to be H-systems though they need not be the L2-system of any group. This paper contains the definitions of an H-system and an L2-system; the proof that an L2-system is an H-system; theorems on idempotents and generation of ideals by idempotents; a functional calculus of self-adjoint elements in an H-system; a simple structure theorem for abelian H-systems; and an example of the L2-system of a special group, which has an interesting structure. Using the theorems of this paper we believe we shall be able to obtain the complete structure of H-systems--the ultimate theorems presumably asserting that every such system is a "direct integral" of simple systems, and that every simple system is a certain type of full matrix systembut with continuous rather than discrete matrices. At present these theorems are not proved but before even considering them carefully it seems necessary to develop the material of this paper as a foundation. One might ask why we choose to investigate this L2-system, in which multiplication is only partially defined, when there are other systems (for example, L1-the integrable functions) which are actually algebras and whose structure is still unknown. Our belief is that in spite of having only a partially defined multiplication our system may be easier to study, in somewhat the same way that certain questions are easier to answer about unbounded hypermaximal operators on Hilbert space than about bounded operators on a general Banach space. That there is a relation between that situation and ours is shown by the fact that left (or right) multiplication by a fixed element in an L2-system is, for many choices of the element, an unbounded hyper

without the assumption of the continuity of the derivative f'(Z) on the closed region R bounded by the curve of integration C, and thereby he has laid deeper foundations for the CAUCHY-RIEMANN theory of fuinctions of the complex variable. An abstract of these memoirs is to be found in the Bulletin of this Society for June, 1899, pp. 427-429. GOURSAT set out by a direct process t to evaluate the integral in question. In the present paper, by an indirect process, I prove that the integral has the value 0. The essential elements of the proof are those of GOURSAT'S first paper; by the modification indicated, and by the imposition on the curve C of a certain condition fulfilled by all the usual curves, one avoids the necessity of introducing the lenmma to which GOURSAT'S second paper is devoted. The necessary preliminary definitions and theorems are given in some detail in ? 1, in which connection I refer especially to JORDAN'S Cours d'Analyse, 2d ed., vol. 1, 1893, and to HURWITZ'S address at the Zurich Congress of 1897 entitled.: Uber die Entwickelung der (illyemeinen Theorie der analytischen Functionen in neuer er Zeit (Verhandlungen des iilathematiker Kongresses inZiirich . .; Teubner, 1898). Then in ?? 2 and 3 I state and prove the two

The matrix A may be chosen to be a Jacobi matrix, that is, Ajj=0 for i-jI >1; and An,n+l> 0 may be required. If I is {0, 1, 2, * }, A is then determined uniquely, assuming the convention that any invariant subspace of A orthogonal to the 0th coordinate subspace will be ignored. (Assuming, that is, that if Ak,k?l =Ak1, k = 0 then in A = ((A ij)) indices will be let run up to k only.) For finite I, A is not in general determined uniquely. This paper gives a convenient canonical form for A. There is little trouble in including in this result the generalized sort of moment problem introduced by Nagy [6], where un above are in 63, the set of bounded self-adjoint operators on a Hilbert space SC. One virtue of the generalized problem is its application to the classical problem; see Proposition 4 below. Accordingly some of the facts outlined in the preceding paragraph may as well be proved for the generalized problem; this is done in ?2, which in fact is essentially a recitation in the wider setting of the proof for the classical case. The main result is in ?3. The following sections mostly examine the classical problem in light of it. I have found the canonical form handy for getting numerical bounds on momenits, but will not discuss this use further. Throughout the paper I use the notation a--1 -a, where a may be a number or an operator. If it is an operator this requires some understanding as to the space on which it operates, but I think I have avoided amlbiguity.

Introduction. Martingale theory for a linearly ordered index set in a finite measure space has been systematically discussed by Doob [4]. As convergence theory and system theory are the two main topics in martingale theory, Helms [6] has extended Doob's results on mean convergence to martingales in a finite measure space, indexed by directed sets, and Krickeberg [8; 9] extended the results of pointwise convergence and stochastic convergence to martingales in a a-finite measure space, indexed by directed sets. Krickeberg uses measure algebra and lattice theoretic methods, and in [8] makes the assumption in pointwise convergence that the martingales have some covering property, which is called Vo in his paper. Dieudonne [3] has given a martingale which is not pointwise convergent and has non-negative, bounded functions in a finite measure space, indexed by a countable directed set. Therefore, some additional conditions like Krickeberg's Vo to the usual ones seem to be necessary for pointwise convergence. However, the condition Vo is too weak to be the only additional condition needed for pointwise convergence of martingales indexed by noncountable directed sets. Some conditions such as terminal separability are needed. For the system theory, Bochner [2] has given some extensions to martingales indexed by directed sets, but without proofs. Some of his results are, unfortunately, not true. Snell [14] has defined regularity (which is closely related to system theorems) of martingales in a finite measure space, indexed by positive integers, and has given some sufficient conditions. He has not discussed the necessary conditions, nor the relation between regularity and the system theorems. In this paper, ?1 gives preliminary definitions and notation. ?2 is devoted to the definition of conditional expectation in a a-finite measure space W, and ?3 to that of martingales indexed by directed sets in W as well as to individual system theorems and inequalities. In ?4, a general pointwise convergence theorem is proved, and a new kind of convergence theorem, based on a differentiation theorem of interval functions [13, p. 192] is given. We

One of the themes of Edwin Hewitt's fundamental and stimulating work [16] is that the Q-spaces (now called realcompact spaces) introduced there, although they are not in general compact, enjoy many attributes similar to those possessed by compact spaces; and that the canonical realcompactification vX associated with a given completely regular Hausdorff space X bears much the same relation to the ring C(X) of real-valued continuous functions on X as does the Stone-Cech compactification gX to the ring C*(X) of bounded elements of C(X). Much of the Gillman-Jerison textbook [12] may be considered to be an amplification of this point. Over and over again the reader is treated to a "/" theorem and then, some pages later, to its "v" analogue. One of the most elegant "/3" theorems whose "v" analogue remains unproved and even unstated is the following, given by Irving Glicksberg in [13] and reproved later by another method in [11] by Zdenek Frolik: For infinite spaces X and Y, the relation 3(X x Y) = gX x / Y holds if and only if X x Y is pseudocompact. The present paper is an outgrowth of the author's unsuccessful attempt to characterize those pairs of spaces (X, Y) for which v(X x Y) = vX x v Y. It is shown (Theorem 2.4) that, barring the existence of measurable cardinals, the relation holds whenever Y is a k-space and vX is locally compact; and more generally (Theorem 4.5) that the relation holds whenever the k-space Y and the locally compact space vX admit no compact subsets of measurable cardinal. The question arises naturally as to when it will occur that vX is locally compact. Our best result in this direction, a part of Theorem 4.8, is, again, not definitive: X is locally pseudocompact if and only if there is a locally compact space Y for which Xc Yc vX. Some of the questions treated here are susceptible to attack when thrown into the uniform space context. See Onuchic [18] and, more intensively, Isbell [17, especially Chapter 8]. I am indebted to Tony Hager for several references, and for allowing me to announce here that, in addition to the results of [15], he has achieved certain new theorems on the relation v(X x Y) = vX x v Y, which being " uniform " and not "topological" have negligible overlap with those of this paper.

Preface. For some time it has been recognized that the properties of symmetry and bounded closure are of cardinal importance in the theory of locally convex spaces. The purpose of this paper is to discuss symmetry and bounded closure with the aid of a new topology, and to discuss the new topology itself. The introduction includes the preliminaries to the rest of the paper. The account of the basic theory of locally convex spaces is fairly complete, but brief. We have given proofs only when they represent some contribution. As the references indicate, most of the theorems of the introduction can be found in the literature. In ?1 we introduce the new topology, called the GF topology, and develop the theory of the GF space. A space is a GF space if and only if it is symmetric and boundedly closed. Study of the GF topology on duals leads to information about interior maps and quotient spaces, information which is not obvious from direct consideration of symmetry and bounded closure, We conclude the section with several examples. In view of the general results of the first section it is important to know what spaces are GF spaces. In ?2 we take up this question for direct products and their duals, and also the questions of symmetry and bounded closure. ?3 serves the same purpose for continuous function spaces. Finally in ?4 we discuss metric and LF spaces and their duals. The influence of the works of G. W. Mackey, J. Dieudonne, and L. Schwartz is apparent throughout the paper, and it is to these authors that we owe our interest in locally convex spaces. The terminology used in the paper is almost exclusively that employed in the works of Dieudonne, Schwartz, and Bourbaki. For the convenience of the reader we list the exceptions. We distinguish between weak and weak-star topologies; our total bounded is the French precompact and corresponds to the definition given in [14]; we owe the term relatively strong to Mackey [13]; where the French authors use filters we make use of nets, as in Kelley [10]; and for a uniform structure topology the natural definition of a Cauchy net leads to a concept of completeness which is equivalent to the French term and stronger than that given in [14]. There are no French equivalents for symmetry and bounded closure.

The first chapter of this paper concerns itself with questions of uniform boundedness of sets of points in a Banach space and sets of functionals on a Banach space, as well as with a group of closely related resonance theorems. A well known example coming under this heading is the theorem of Toeplitz [36] t stating that supm Z I= a,.I < oo providing 77m =Zn-, -amntn converges whenever S, does. Another is the theorem of Hahn [12 ] stating that if an arbitrary continuous function has the partial sums of its Fourier expansion, with respect to an orthonormal sequence of bounded functions (n, essentially bounded, then the sequence f ZX= 1c^,=(x)co,(t) I dt is also essentially bounded. Still another is the theorem stating that if the adjoint of an everywhere defined transformation between Banach spaces is everywhere defined, then the transformation is continuous. This was proved, at least for Hilbert space, by von Neumann [21 ], Stone [34 ], Tamarkin ( [34], p. iv), and Stone and Tamarkin [35 ] and is probably not usually thought of as a theorem on uniform boundedness. Most of the results w-e have in the first chapter are new, others have been proved only in special cases, and some are well known but the proofs heretofore given have been different. Previous methods for discussing questions of uniform boundedness divide themselves into three groups (i) those associated with the names of Lebesgue [18], Banach [2], Hahn [11], and Hildebrandt [13 ]; (ii) those characterized by the elegant and direct use of the Baire category theorem as in the works of Banach [1], Saks [31], Saks and Tamarkin [32], and others; and (iii) those employed in a recent theorem of Gelfand [9] the proof of which is closely related to that of the category theorem. The second chapter of this paper is concerned with cases where a weak limiting process implies a strong one. These cases seem to be rather rare but probably more will come to light in the future. The theorem quoted above, stating that the existence of the adjoint implies the continuity of the function, belongs to the class of questions discussed in Chapter II as well as those discussed in Chapter I, while its analogue (Theorem 42) in the Boolean ring of

Introduction. By "the space of continuous functioins" we mean the Banach space C of real continuous functions on a compact Hausdorff space X. C, its first dual, and its second dual are not only Banach spaces, but also vector lattices, and this aspect of them plays a central role in our treatment. In ??1-3, we collect the properties of vector lattices which we need in the paper. In ?4 we add some properties of the first dual of C-the space of Radon measures on X-denoted by L in the present paper. In ?5 we imbed C in its second dual, denoted by M, and then identify the space of all bounded real functions on X with a quotient space of M, in fact with a topological direct summand. Thus each bounded function on X represents an entire class of elements of M. The value of an element f of M on an element ,i of L is denoted as usual by f(,u). If f lies in C then of course f(,u) =,u(f) (usually written f fd,u), and thus the multiplication between M and L is an extension of the multiplication between L and C. Now in integration theory, for each ,uEL, there is a standard "extension" of ffd,u to certain of the bounded functions on X (we confine ourselves to bounded functions in this paper), which are consequently called ,i-integrable. The question arises: how is this "extension" related to the multiplication between M and L. ??6-8 (and ?12) are devoted to this question. We show that given a bounded function f on X the equivalence class of M represented by f contains two distinguished elements, which we denote by f* and f*, such that for every Radon measure ,u, f*(,u) = f*fd,u and f*(,u) = f*fdM, where f* and f* are the lower and upper integrals of f with respect to ,u. Moreover, if f is integrable for every Radon measure, then f* and f* coincide, and we have a unique element whose value on each ,u is ffd4. In ?9 we study order-convergence in M. It is in this area that Af offers its most striking immediate gain in clarity over consideration of only the bounded functions oil X. We cite two examples. The first example: The characteristic functions of finite subsets of the real initerval 0< x < 1 form a directed system which order-converges to the constant function 1, while their Lebesgue integrals, being all zero, do not converge to the Lebesgue integral of 1. In M, however, this directed system order-converges to an element different from 1, and the above unsatisfactory situation disappears. In point of fact, order-convergence is perfectly well-behaved in M: if a directed system order converges, its values on every Radon measure converge

1. A mean on a semi-group 2 is a positive linear functional of norm one on the space m(z) of bounded, real-valued functions on 1. A bounded semigroup S of linear operators from a Banach space B to itself is called ergodic if there exists a system cd of averages A such that for every S in s limA (AS-A)=limA (SA -A) =0; we have three strengths of ergodicity of S according as uniform, strong, or weak convergence is used in the operator algebra. The first part of this paper deals with the relationship between existence of invariant means and ergodicity of bounded representations. In Theorem 2 it is shown that weak ergodicity of every bounded representation of I is equivalent to weak ergodicity of the right and left representations of f by right and left translations on m(2), and equivalent to the existence of a mean on m(z) invariant under right and left translations. These conditions, in turn, are equivalent to existence of a directed system of finite means on mn(z) converging weakly to two-sided invariance under all right and left translations of m(z). Uniform ergodicity is similarly related to existence of finite means converging in the norm of m(z2)* to two-sided invariance (Theorem 4). The second part of this paper gives some sufficient conditions for existence of invariant means or of finite means converging in norm to invariance. For the former, the Markoff method of proof by fixed-point arguments is applied (?5) to "solvable" semi-groups and groups, and to semi-groups which are the union of expanding directed systems of sub-semi-groups with means. (For example, a group G such that every finite subset generates a finite subgroup has an invariant mean.) It is also proved by a direct construction (Theorem 6) that if G is a normal subgroup of a group H and if G and H/G have two-sided invariant means, so has H. In ?6 a parallel result is proved for finite means converging in norm to invariance. These results greatly increase the family of groups known to have invariant means. Solvable groups formed the only such class previously known; ?6 now shows that a solvable group satisfies a stronger property; it has a system of finite means converging in norm to invariance.

It is shown here that the mixed initial-boundary value problem for the Euler equations for ideal flow in bounded domains of R3 has a unique solution for a small time interval. The existence of a solution is shown by converting the equations to an equivalent system involving the vorticity and applying Schauder's fixed point theorem to an appropriate mapping. Introduction. The purpose of this paper is to prove the existence and uniqueness of classical solutions to the mixed initial-boundary value problem for the Euler equations for an ideal incompressible fluid in bounded domains of R3 for a small time interval. This has been shown by Ebin and Marsden [3] for compact Riemannian manifolds possibly with boundary. In this paper we obtain a new proof by converting the equations to an equivalent system involving the vorticity and solving this by using the Schauder fixed point theorem. The technique is similar to that of Kato [4] where he proved existence of ideal flow in bounded domains of the plane for an arbitrary time interval. The author also used a similar method to obtain an existence result for ideal flow in all R3 by showing that a solution is the limit of Navier-Stokes flow as the viscosity goes to zero [7]. Existence of a solution for a short time in all R3 was originally shown by Lichtenstein [5]. The Euler equations for ideal incompressible flow in a domain D in R3 with boundary bD are (Et) ar/at + (r grad)r =-grad P + B, V * U = O, with constraints U n = 0 on bD and r(0) = A, where x is a point in D U bD; t E [O, T], Iv(X, t) is the velocity vector, P(x, t) is the (scalar) pressure, B(x, t) is the external force field vector, n(x) is the outward normal vector to bD and A(x) is the initial velocity vector. By formally computing the curl of (E') we get the system Presented to the Society, January 18, 1972 under the title The existence and uniqeness of nonstationary idealflow in bounded domains in R3; received by the editors October 12, 1971 and, in revised form, March 20, 1972. A J1S (MOS) subject classifications (1970). Primary 35F25, 35F30, 35Q05, 76C05.

Introduction. The first proof of uniqueness of a plane subsonic flow of a compressible fluid past an obstacle was given by Bers [1]. This proof utilizes an elaborate mathematical apparatus encompassing some of the most advanced tools of modern function theory. A conceptually simpler proof under somewhat weaker hypotheses and a proof of the Joukowsky force formula, due to the authors [3], make essential use of a general existence theorem and hence cannot properly be called elementary. In this note we show that the uniqueness of a compressible flow and the Joukowsky force formula can be obtained directly from a simple geometrical property of the velocity field defined by the flow. Specifically, we base our proof on the facts that the velocity components u, v of the flow define a quasi-conformal(') mapping of the (x, y) plane, and that a quasi-conformal mapping with dilatation ratio not exceeding K = 1/M satisfies at each point a Hilder condition with exponent ,. Until recently we would have considered this proof more difficult than our original one. We are, however, now able to refer to the preceding paper [5 ], in which a simple demonstration is given for the needed lemma. We note, further, that a simpler form of the arguments in [5] suffices for the present paper. The problem of proving uniqueness of a flow past an obstacle P can be divided into two parts; to prove the uniqueness if the velocity at infinity and circulation are both prescribed, and to prove that if P has a corner or cusp T the circulation is uniquely determined by the condition that the speed is finite at T. We shall settle both points here, but for the second we need the additional assumption, not used in [1] or [3], that the velocity components have bounded derivatives up to P. (This assumption can be avoided by a discussion analogous to that in [3, ?6.2].) By the same method we shall also obtain a new proof, simpler than that of [3], for the force formula of gas dynamics. 1. Uniqueness with prescribed circulation. As in [3 ] we define the velocity potential of a flow past P to be a solution k(x, y) of

This paper adapts the method of F. Riesz to the proof of certain general ergodic theorems for Abelian semi-groups of operators on a Banach space to itself. The main features of the method are that no measurability conditions are imposed on the semi-group under consideration and that consistent use of the second conjugate space and its compactness properties make it possible to replace the compactness conditions often imposed by a more natural restriction on the transforms of points. Theorem 1 and the various supplementary results include as special cases theorems of Lorch [10], Dunford [7], Yosida [15], F. Riesz [12], and Cohen [6]. It overlaps the work of Alaoglu and Birkhoff [3] at those points where they consider Abelian cases; for example, Corollary 8 is a great generalization of their Theorem 5. Section 1 contains some introductory material on conjugate spaces and adjoint operations. Section 2 introduces bounded Abelian semi-groups of operators and near invariance of a system of set functions on such a semi-group; this section also contains the principal theorem (Theorem 1) of the paper. The form of this theorem raises three questions ((A) to (C) at the beginning of ?3). The answer to (A) shows, among other things, that every Abelian semi-group has a property much like "ergodicity" in the sense of Alaoglu and Birkhoff; Theorem 3 is the main result here. The answer to (B) again indicates the importance of reflexivity in theorems of this type; Corollary 8 is one example. Two special cases of (C) give a generalization of Dunford's theorem (Theorem 5) and a theorem on bounded Abelian semi-groups of projections (Theorem 6) which has not, so far as I know, been considered before. 1. Some properties of Banach spaces. If B is a Banach space(2), let B* be the set of all linear-that is, additive and continuous-real-valued functions on B. If, for A in B*, |I|3 =SUpjjbJJ 0, k a positive integer, and I1, . , f3k in B*. With this topology B is a linear topological

Introduction. This paper is mainly concerned with the characterization problem for bounded spectral operators (see [5]). We restrict the discussion to scalar type operators with real spectrum, which we call pseudo-hermitian operators (p. h.). The criteria are given in terms of certain properties of the exponential group generated by the operator. The simplest criterion of this kind is for Hilbert space: a bounded linear operator S is p.h. if and only if || eitS| ? M < oo for all real t. This statement is false for Banach spaces, and even for reflexive Banach spaces. In the latter case, we give additional necessary conditions on the group eitS, which together with its uniform boundedness are both necessary and sufficient for S to be p.h. Another useful criterion is the following: a bounded linear operator S on a reflexive Banach space is p.h. if and only if for everyf in L1(R) (R the real line), we have II fRf(t)e itsdt || _ M || f l,,, where the norm on the left is the operator norm, f is the Fourier transform off and || * is the sup norm. Unlike the general theorems in [5] about the spectrality of an operator, the characterizations obtained in this paper are easily applied to analytically given operators on concrete Banach spaces. An example is discussed in ?4. Applications are discussed at the end of ?5. ?2 deals with an improvement of a theorem of Foguel [7] about the resolutions of the identity of sums and products of commuting spectral operators. ?1 contains preliminaries. The characterization problem for spectral operators is discussed in ??3 and 5.

Several authors have given definitions of an end point of a continuum, making use of properties which are possessed by an end point of a straight line interval, but which are not possessed by any interior point of the interval. We shall show in the present paper that the properties used in these so-called "definitions" are not logically equivalent, and shall determine the logical relations which do exist among these properties under various conditions. In Part 2, we shall show the equivalence of a number of these properties in the case of a continuous curve. In Part 3 is shown the equivalence with the first set, of two additional properties in the case of a continuous curve of a special type. In Parts 4 and 5, we determine the logical relations existing among certain of these properties in the case of a bounded continuum. Finally, in Part 6, some theorems are proved concerning points having one or more of the given properties. In this paper we shall consider only plane point sets, although in a number of cases it is obvious that our results are true in space of any number of dimensions. In regard to the use of the word "end point," we intend hereafter to use this word only in the sense of a point of a continuous curve satisfying Wilder's definition or one of the other definitions equivalent to it, in other words, a point having any one of the properties 1-7 given in Part 2. We shall not use the word "end point" in referring to a point of a continuum which is not a continuous curve. The examples of Part 4 show that a point of a continuum may have certain of the given properties and yet be so placed with respect to that continuum as hardly to deserve the name of "end point."

on the range space, the class of semigroups which permit the two types of invariant extensions is contained in the class of functions which have invariant means definable on the associated Banach space of bounded, real-valued functions defined on these semigroups. Further it was shown that every semigroup known to have an invariant mean also permitted the two types of extensions. When structure in addition to the minimal required structure is placed on the range space, e.g., when the space is an ordered subspace of the conjugate space of an ordered linear space with reproducing cone, a semi-group which has an invariant mean also has the two extension properties relative to these particular range spaces. Many of the standard function spaces satisfy these conditions. In addition this paper will exhibit conditions which guarantee the continuity of these extensions in the case when linear topological spaces are considered and make some application to real-valued functions. ?11 outlines definitions and results preceding this paper. ?III contains theorems on the existence of invariant extensions for a given semigroup G with an invariant mean and a given range space V. Various extra conditions added to the known necessary condition that V be a boundedly complete vector lattice are sufficient for these extensions. Some converse results are also given under which the existence of some sort of invariant extension implies that G has an invariant mean. ?IV contains examples of spaces satisfying the conditions imposed on the space V. ?V discusses continuity of the extensions when order and topology are both present. ?VI presents applications of the theorems of ??III and V to some of the examples of ?IV.

Introduction. This paper is concerned with extremal problems in the family of bounded analytic functions in a multiply-connected domain D, and it is concerned with extremal problems for the mean modulus f cIf Ids of meromorphic functions f in D taken over the boundary C of D. These two types of problems are shown to be closely related, and solutions are obtained simultaneously for both types by a method of contour integration. An altogether analogous method was exploited by Grunsky in his thesis [8](2) for the investigation of schlicht functions in a multiply-connected domain. Thus it is interesting to remark that the fundamental distortion theorems of schlicht conformal mapping theory have been developed by Grunsky by a method which we are able to apply here to obtain the fundamental distortion theorems for bounded functions. It will appear, then, that the generalization of Schwarz's lemma to multiply-connected domains and the generalization of the Koebe distortion theorem can be carried out by a unified technique(3). We shall find in addition to this that, while the recent papers of Bergman and Schiffer [5, 16] have developed a close relationship between the theory of schlicht canonical mappings and the Bergman kernel function [3], we are able to develop here a relationship between the theory of bounded functions and the Szego kernel function [19]. Thus the Szego kernel function does for the theory of distortion of bounded functions what the Bergman kernel function does for the theory of distortion of schlicht functions. We point out that both these kernel functions are actually differentials, and that in the Szego case one is dealing with length and in the Bergman case one is dealing with area. Thus the mean modulus fc f I ds, or ffD IfI 2dxdy, should be thought of as a length, or area, and not as a mean modulus. All these remarkable relationships are brought to light by using the simple boundary relations satisfied by the classical domain functions, such as

The two main results in this paper are analogues of the Stone-Weierstrass theorem for real-valued functions, obtained by using different function space topologies. The first (Theorem 2.3) is a Stone-Weierstrass theorem for unbounded functions. The second (Theorem 3.6) is a theorem for bounded functions; it is stronger than the usual theorem because the topology is larger than the uniform topology. In fact it is the strongest possible theorem in a sense made precise in Theorem 3.5. However, the usefulness of the result is, in both cases, limited by the fact that the topologies used are far from being as well behaved as the uniform topology. Some specific defects of these topologies are discussed in Proposition 2.4, Proposition 3.9, and Example 3.10. (The topologies are defined in ?1.) The formulation of the Stone-Weierstrass theorem on which the analogues in the present paper are based is the following: If X is a completely regular topological space, then X is compact iff every strongly separating subalgebra of C*(X) is u-dense, where u denotes the uniform topology. In [5] Lorch has studied a more general class of function algebras than those of the form C*(X). All of the results for bounded functions in this paper are formulated in this more general setting (see ?1).

The present article is intended as a survey of the title subject. The material is closely intertwined with all branches of harmonic analysis. Historically, the rigorous foundations of the theory arise in Riemann's treatment of trigonometric series, and spectral theory is essentially equivalent to the study of formal multiplication. The original motivation for the modern treatment, due to Wiener, Carleman, and Beurling, came from the study of integral equations with convolution kernels. There are applications to linear partial differential equations with constant coefficients, and there is a very close connection with problems about entire functions bounded on a line. My own concern with the topic commenced with questions about the theory of approximation for multiple Fourier transforms. When I first looked at the subject the basic material did not appear very well organized, and certain elementary facts were not recognized as immediately obvious to experts in the field. Thus, in 1954, I set about to record what was known with certain useful additions. Shortly after the original version of this work was finished in 1956, the paper of DOMAR appeared. (An author's name in small capitals indicates a reference to the bibliography.) There was a considerable overlap, for, although the two papers were quite independent, both were heavily influenced by unpublished notes of Beurling. I have revised the paper in an attempt to suppress details which may be found elsewhere. Also, I have taken advantage of more recent work by others and myself to improve the material of the last half of the paper. After a preliminary section introducing much of the notation and basic definitiong, the contents of this paper are divided into six parts. The headings are: 1. The spectrum. 2. The point spectrum. 3. Potential theory and spectral analysis. 4. The spectral synthesis problem. 5. Representations. 6. Examples of spectral synthesis.

In Part I of this paper we construct a bounded-truth-table-complete set which is not creative, and in Part II we construct an infinite and coinfinite recursively enumerable (r.e.) splinter which is not a cylinder. Although these results appear unrelated, the proofs given here are similar and so have been included in a single paper. The two parts may be read independently of one another. Each of these results yields as corollary the existence of pseudo-creative sets which are not cylinders. In fact, in Part II we construct a pseudo-creative set S for which there is no total recursive function f such that x C S implies f (x) C S-{ x } and x C S' implies f (x) C S'-{ x } .

The Chaplygin equation, which arises in the study of transonic flow, is one of the simplest equations of mixed type. It was shown that with appropriate hypotheses on the function h a problem of this type could be stabilized (i.e., the solution made to depend continuously on the data) by restricting the solution to lie in the class of functions whose L2 integrals are bounded by some prescribed constant M. Similar restrictions have previously been shown (see e.g., [2], [7]) to guarantee continuous dependence in certain improperly posed problems for elliptic and parabolic equations. However, little study has up to now been devoted to improperly posed problems for equations of mixed type. For that matter one is rarely able to determine just what is a well set problem for such an equation. In this paper we demonstrate that the convexity arguments which led to stability inequalities for the Chaplygin equation [6] actually can be carried over with some modification to yield stability inequalities and error bounds in improperly posed problems for a wide class of quasilinear partial differential equations of mixed type. We shall be concerned here only with the question of obtaining such inequalities. We leave aside the difficult question of existence. The particular equation which we consider is by no means the most general one amenable to the convexity method. We restrict our attention to a relatively simple equation but one which exhibits most of the troublesome features which would appear in the more general operators. A survey of the literature on improperly posed problems will appear in a forthcoming paper by Payne [5]. In this paper we consider the following initial-boundary value problem for a cylindrical domain Q = D x (0, Y), where D is an arbitrary n-dimensional domain with smooth boundary D

Part I of this paper has to do with connected sets of cut pointst of a given continuous curve. It is shown that, in the plane, any two points belonging to a connected set K of cut points of a given continuous curve M lie together in an arc which is common to K and a point set consisting of the boundaries of a finite number of complementary domains of M. G. T. Whyburnt calls attention to the fact that from his results it follows that K is arcwise connected. To show that any two points of K can be joined by an arc which is common to K and a point set consisting of the boundaries of a finite number of complementary domains of M, is the object of Part I. Part II has to do with a totally disconnected closed subset K of a given plane continuous curve M no subset of which disconnects M. R. L. Moore? has shown that, in the plane, any two points not belonging to a bounded continuous curve can be joined by an arc which does not disconnect the continuous curve. The object of Part II of this paper is to show that if, in the plane, M is a bounded continuous curve which contains no domain, and K is a closed and totally disconnected subset of Al, such that no subset of K disconnects M, then there exists an acyclic continuous curve T, containing K, such that (1) all the end pointsll of T belong to K and (2) the point set M T is totally disconnected and M-T is connected. I wish to acknowledge my indebtedness to Professor R. L. Moore, not only for the suggestions of the problems treated, but also for his many helpful crltici;crn ;P their solutions. To him is due, in a large measure, the success

THnORUM I. If the points z1, Z2, Z3 vary independently and have circular regions as their respective loci, then the locus of the point Z4 defined by the real constant cross ratio X=(Zl, Z2, Z3, Z4) is also a circular region. It is the purpose of the present paper to consider generalizations of and other results related to Theorem I, primarily the determination of the locus of the point Z4 defined as in Theorem I, when zi, Z2, Z3 vary independently so as to have certain prescribed loci. Thus one may raise the following question: If two variable points lie respectively on two fixed circles, where does the mid-point of their segment lie? The answer is contained in Theorem VI. Or again, if the loci of the points zi, Z2, Z3 are regions each bounded by a number of circles, what can be said of the locus of z4? The answer is given by Theorem VII. All the results proved concerning such loci as these can be interpreted in terms of the roots of the jacobian of two binary forms. Such an interpretation is given in ?12. The chief method used in the present paper to determine the locus of Z4 in any given case is that suggested in I (pp. 102, 103, footnote), namely the determination of the locus of z4 when z1 and Z2 are held fast and Z3 varies over its locus; the determination of the locus of this locus of z4 when z1 is kept fixed but Z2 varies over its locus; and finally the determination of the locus of this new locus of Z4 when z1 varies over its prescribed locus. In the present paper this method is carried through geometrically. The same method has been used by Professor A. B. Coblet to determine analytically the locus of Z4 in Theorem I

In this paper we shall confine ourselves to a two-dimensional euclidean space which we shall denote by the symbol S. R. L. Mooret has given the following extension of a theorem by Zoretti:t If K is a bounded maximal connected subset of a closed point set M and does not separate space, and e is a positive number, then there exists a simple closed curve containing no point of M, such that the interior of this curve contains K but contains no point whose distance from K is greater than e. Zoretti's conclusion is weaker than Moore's in that it says nothing about the points on the interior of the curve surrounding K. On the other hand, Moore finds it necessary to use a stronger hypothesis than Zoretti's; namely that K does not separate the plane. In this paper we consider among other things the question of the conditions involved in a combination of Zoretti's hypothesis and Moore's conclusion, and show in Theorem I that a result analogous to

for almost all t. In this definition it is assumed, of course, that the integral on the right-hand side of (1.2) is in the domain of A. In a recent paper [6], Friedman and Shinbrot have studied the equation (1.1) even in the more general case where xo and A depend on t and -, respectively. They proved theorems of existence, uniqueness, differentiability and asymptotic behavior of solutions. They also constructed fundamental solutions and derived asymptotic bounds for them. We recall [6] that if xo is in the domain of A4, for some ,u>0, then the solution of (1.1) is given by S(t)xo. The purpose of the present paper is to derive monotonicity theorems for solutions of (1.1). We shall generalize some of the monotonicity theorems of Friedman [2] (see also [4]) from the case X= R1 (R1 the one-dimensional Euclidean space) to the case where X is any Banach space. In ?2 we give some auxiliary results. These results are concerned with Volterra equations in one-dimension (i.e., X=R1). In particular, we study the behavior of the solutions with respect to a certain parameter. In ?3 we give an integral formula for S(t) in case A is a bounded operator. For A unbounded, we construct a fundamental solution as a limit of fundamental solutions corresponding to the bounded operators A(I+ A/n) -1. We prove that the fundamental solution coincides with the fundamental solution of [6, Chapter 1]

1.1. Inherent in the definitions of one of the basic concepts in modern theory of the area of a continuous surface (see bibliography) there is a question which this note intends to resolve. The situation will be described presently. Since the question first comes to light in the first two chapters of the paper by Rado and Reichelderfer titled A theory of absolutely continuous transformations in the plane-hereinafter cited as R2 [1] (1)-the reader is requested to consult that paper freely while reading this note. Definitions, theorems, and-as far as practicable-notations in R2 [1 ] will be used in the sequel. 1.2. Let S denote a bounded point set in the uv-plane. Given a pair of finite-valued, real-valued functions x(u, v), y(u, v), defined, bounded, and continuous on S, these determine a bounded continuous transformation T:x=x(u, v), y=y(u, v), (u, v)ES, from the set S into the xy-plane (cf. R2 [1, 1.8 ]). For conciseness, complex notation z = x+iy, w = u+iv is introduced, and T is written in the form T:z=t(w), wCS, where t(w) =x(u, v)+iy(u, v), u+iv=wES, is finite-valued, complex-valued, bounded, continuous on S. If E be any set in the w-plane, the transformation T associates with each point g in the z-plane a crude multiplicity with respect to E, N(a, T, E), defined to be the number (possibly + co) of distinct points o in the set E T-F(g), where T-1(g) denotes the set of all points o in S for which t(W) = (cf. R2 [1, 1.10, 1.8]). 1.3. Using the crude multiplicity, Banach [1] and Schauder [1] have developed a beautiful, though restrictive, theory of continuous transformations, which has been completed in many respects by Rado [2 ]. But the crude multiplicity has been found to be generally too large to play an effective role in modern theory of the area. A study of the works of Ge6cze led Rado to the idea of replacing the crude multiplicity by a generally smaller multiplicity (see Rado [1]), which has come to be known as the essential multiplicity. 1.4. The intuitive idea of the essential multiplicity is embodied in the following statement. DEFINITION. Let T:z=t(w), wES, be a bounded continuous transforma

If ? is a function whose domain is a subset E of the set of curvilinear convergence off, then ? is called a boundary function for f if, and only if, for each x E E there exists an arc y at x such thatf(z) -+ ;(x) as z -* x along y. Let S be another metric space. We shall say that a function ? is of Baire class < 1(S, M) if (i) domain ? = S, (ii) range ? c M, and (iii) there exists a sequence {0n} of continuous functions, each mapping S into M, such that n, -* 0 pointwise on S. We shall say that ? is of honorary Baire class < 2(S, M) if (i) domain ? = S, (ii) range ? c M, and (iii) there exists a countable set Nc S and there exists a function b of Baire class _ 1(S, M) such that ?(x) = +(x) for every x E S-N. It is known that iff is a continuous function mapping D into the Riemann sphere, then the set of curvilinear convergence off is of type Fa,, and any boundary function forf is of honorary Baire class _ 2(C, Riemann sphere). (See [3], [4], [5], [6], [9].) J. E. McMillan [6] posed the following problem. If A is a given set in C of type F,6, and if ? is a function of honorary Baire class _2(A, Riemann sphere), does there always exist a continuous function f mapping D into the Riemann sphere such that A is the set of curvilinear convergence of f and ? is a boundary function for f? The purpose of this paper is to give an affirmative answer to McMillan's question. However, the corresponding question for real-valued functions remains open. (See Problems 1 and 2 at the end of this paper.) In proving our result, we first give a proof under the assumption that ? is a bounded complexvalued function, and we then use a certain device to transfer the theorem to the Riemann sphere. As we shall indicate in an appendix, the same device can be

Introduction. In the study of uniform approximation to a function of a complex variable by polynomials or by bounded analytic functions, Lipschitz conditions have proved extremely useful in relating degree of approximation to continuity properties of the functions approximated. Parts of this theory are analogous to the older study (S. Bernstein, D. Jackson, de la Vallee Poussin, Montel) of approximation to real periodic functions by trigonometric sums, where Lipschitz conditions have also proved useful. Hardy and Littlewood [2, p. 633 ](2) first pointed out that trigonometric approximation in the mean is likewise closely related to integrated Lipschitz conditions; proofs of the theorems stated by Hardy and Littlewood were first published by Quade [l]. The object of the present note is to indicate rapidly and with a minimum of detail, that degree of approximation in the mean by polynomials in the complex variable and by bounded analytic functions is also conveniently investigated by use of integrated Lipschitz conditions. The investigation leads naturally to the use also of classes of analytic functions satisfying integrated Zygmund and integral asymptotic conditions. We shall approximate functions on an analytic Jordan curve or set of curves in the z-plane. Throughout the paper 7 denotes the unit circle, C a single analytic Jordan curve or a finite number X of mutually exterior analytic Jordan curves Cy, j=l, 2, • • , X; if X = l, Ci = C. A function z = Xi(w), w = reie, maps the closed interior of y: \w\ =1 onto the closed interior of Cy one-to-one and conformally; w = fl,(z) denotes the function inverse to z = Xy(-0; Cy denotes the interior of Cy together with Cy. Similarly, w = cp(z) maps the exterior of C conformally (not necessarily one-to-one) onto | w\ > 1 so that 00 = 1) is the image in the exterior of C ol \w\ =R under this mapping. In this paper, the letter s will denote arc-length measured on C, the letter p a number not less than unity, the letter k a non-negative integer unless otherwise indicated, the letters M and L with or without subscript constants independent of re, z and w. A function f(z) belongs to the class Hp on y if it is analytic interior to y and if Mp[f(rew)}={fln\f(reiB)Yd6}1'r> is bounded for 0

In this paper we obtain a classification of motion for Newtonian gravitational systems as time approaches infinity. The basic assumption is that the motion survives long enough to be studied, i.e., the solution exists in the interval (0, co). From this classification it is possible to obtain a sketch of the evolving Newtonian universe. The mathematical study of Newtonian gravitational systems has a long history and has inspired a considerable amount of modern mathematics such as, among other topics, ergodic theory, algebraic topology, qualitative theory of differential equations and some functional analysis. Yet very little seems to be known about gravitational systems beyond the two-body problem. In 1922, J. Chazy [1] was able to classify the motion of the three-body problem as time, t, approaches infinity. In 1967, H. Pollard [7] obtained the first general n-body results as t -+ oo. He obtained results concerning the maximum and minimum spacing between particles as t -+ oo. His work suggests that the behavior of systems with nonnegative energy is in some sense a generalization of the twoand three-body problems. It is the purpose of this paper to sharpen these results and to provide the first classification of motion of the n-body problem, as t -+ oo independent of the sign of the energy. With this classification of motion a sketch of the evolution of Newton's universe as t -+ oo is possible. Also several remaining problems on the growth of systems are partially answered. It is interesting to note that previous classifications of motion have been attempted in terms of the sign of the total energy of the system. It turns out that this approach is far too restrictive and that the classification should be made according to the rate of separation of the particles, as is done here. It will be shown that in the absence of motion that we will call oscillatory and pulsating, the n-body problem is quite well behaved. It separates into clusters where the mutual distances between particles are bounded as t -+ oo. The clusters form subsystems characterized by the separation of clusters like t213. The centers of mass of the subsystems separate asymptotically from each other as Ct. Most of the results depend quite heavily on Tauberian theorems of the type of Landau [20, p. 194] (we follow the customary usage of the o-O symbols): Presented to the Society, January 23, 1970 under the title Classification of motion for gravitational systems; received by the editors March 12, 1970 and, in revised form, June 19, 1970. AMS 1969 suibject classifications. Primary 7034; Secondary 3440, 85XX.

Let D be the open unit disk in the complex plane and let C be its boundary, the unit circle. If 4 E C, then by an arc at t we mean a simple arc y with one endpoint at 4 such that y { } ( D. In this paper we use the term boundazy function in the following sense. Iff is a function defined in D and b is a function defined on a set Sc: C, then we say that b is a boundary function forf if, and only if, for each 4 E S there exists an arc y at * such that f(z) approaches 0(a) as z approaches * along y. It is known that if b is a boundary function for a continuous function, then b can be made into a function of Baire class 0 or of Baire class 1 by changing its values on at most a countable set of points [4, Theorems 2, 3], [6, Theorem 6], [8, Theorem 3]. Hence b is of Baire class at most 2. Conversely, if b is a function defined on C such that b can be made into a function of Baire class 0 or 1 by changing its values on at most a countable set, then b is a boundary function for some continuous function [1, Theorem 8]. Bagemihl and Piranian gave an example [1, Theorem 6] of a harmonic function having a boundary function defined on C that is not of Baire class 0 or 1, and they asked [1, Problem 5] whether there exists a bounded harmonic function having a boundary function defined on C that is not of Baire class 0 or 1. In the present paper we answer this question by constructing the desired function. We then show that, despite this example, a boundary function for a bounded harmonic function always resembles a function of Baire class 0 or 1 in this respect: its set of discontinuity points is of the first category. We say that a functionfdefined in D has the asymptotic value a at a point ' E C if there exists an arc y at 4 such that f(z) approaches a as z approaches * along y. We say that f has general limit a at 4 if f(z) approaches a as z approaches 4 with no restrictions other than that z E D. Let Pr(6) denote the Poisson kernel; that is

0. A classical result of Fatou states that if u(z) is harmonic and bounded in lzl < 1 then u has a nontangential limit at almost every point ei?. The same conclusion holds if u is only bounded from below. These results have a local analogue -namely, if u(z) is harmonic in lzl < 1 and at each point ei0 of a measurable set E there is some cone in which u(z) is bounded from either above or below then u(z) has a nontangential limit at almost every eiO & E. With the aid of conformal mapping one can show the above results hold for regions of the plane more general than lzl < 1. Methods similar to those used for lzl < 1 may be applied to functions which are harmonic in the unit ball of Euclidean (n + 1)-space. However, since we lack a conformal mapping theorem for more general domains DcE +1 the situation there is more technically complicated. Known results for certain kinds of domains D c E, + 1 are due to Brelot and Doob [2], Calderon [3], Carleson [4] and Widman [7], and the purpose of this paper is to obtain nontangential boundary values for functions which are harmonic in still more general types of domains D' En + 1 and bounded from above or below in cones. The domains which we consider are all regular domains for the solution of the Dirichlet problem. For a regular domain D, our result may be stated as follows. Let EC aD and suppose for each Q E E there is a cone with vertex Q which is exterior to D. Then if u is harmonic in D and is bounded from above or below in a cone with vertex Q for each Q E E, u has a finite nontangential limit at each Q E E except for a set of harmonic measure zero. It is natural that the exceptional set be one of harmonic measure zero since if Ec AD is any set of harmonic measure zero it is easy to construct a positive harmonic function with boundary value +oo at each point of E. Our main result is a consequence of the basic result that if u is harmonic and bounded in a starlike Lipschitz domain D, then u has a finite nontangential limit at each Q E AD except for a set of harmonic measure zero. In ?1 of this paper we define the terms starlike Lipschitz domain, nontangential limit, and harmonic measure. We also include some elementary consequences of these definitions and some essential theorems on the differentiation of integrals.

Let X be a real Banach space, D a bounded open subset of X, and D ¯ \bar D the closure of D. In §1 of this paper we establish a general fixed point theorem (see Theorem 1 below) for 1-set-contractions and 1-ball-contractions T : D ¯ → X T:\bar D \to X under very mild conditions on T. In addition to classical fixed point theorems of Schauder, Leray and Schauder, Rothe, Kransnoselsky, Altman, and others for T compact, Theorem 1 includes as special cases the earlier theorem of Darbo as well as the more recent theorems of Sadovsky, Nussbaum, Petryshyn, and others (see §1 for further contributions and details) for T k-set-contractive with k > 1 k > 1 , condensing, and 1-set-contractive. In §§2, 3, 4, and 5 of this paper Theorem 1 is used to deduce a number of known, as well as some new, fixed point theorems for various special classes of mappings (e.g. mappings of contractive type with compact or completely continuous perturbations, mappings of semicontractive type introduced by Browder, mappings of pseudo-contractive type, etc.) which have been recently extensively studied by a number of authors and, in particular, by Browder, Krasnoselsky, Kirk, and others (see §1 for details)

In this paper we are concerned with the convergence of solutions of elliptic and analytic Monge-Ampere equations. Theorem 1 gives the principal result of this paper. The example on p. 372 indicates the possibility of certain types of singularities which for linear elliptic equations cannot occur. Theorems 2 and 3 give sufficient conditions for the analyticity of the limit function. These conditions allow applications to certain problems of the differential geometry in the large. Our method consists in introducing a regularizing contact transformation which transforms convex functions into functions with bounded second derivatives and thus makes possible the reduction of Theorem 1 to the principal result of the first part of this paper. 1. Regularizing contact transformation. Consider the contact transformation T of an (x, y, z)-space into a (%, -q, )-space generated by the relation

where x = (x1, * , xn) C Rn, bj(x), j = 1, * , n and V(x) are real-valued functions on Rn, aj = d/ (dx1), i = V 1 and CZ Z), a dense linear subset of L2(Rn). i) is called the domain of A (b; V; t) and is denoted by t (A). The domain ?) and/or (b; V) will be omitted from A(b; V; Z) when no confusion should arise. In practice, the spectral resolution of a certain self-adjoint extension, A, of A is important. One important technique used in studying pertinent properties of the spectral resolution of A(b; V;Z) is to represent e Q-L2(Rn), by the Wiener integral of a certain functional F(b; V; O) and then to translate the (at least formally) transparent properties of this integral into less transparent but important properties of A. For example, see [14] and [ 18]. This functional integral representation has been shown to be rigorously valid for the case where bj(x) = 0 (j = 1, * *,*n), and where V(x) is Borel measurable and bounded from below. Unfortunately, A is only given implicitly. See [ 8], [ 9], and [ 10] . It is the purpose of this paper to extend the validity of the functional integral representation of e-At% to the case where b(x) = (b1(x) , .., bn(x)) is a sufficiently smooth vector field which doesn't grow too rapidly at infinity, and where V(x) can have certain negative singularities. Moreover, A will be given explicitly(). The Schrodinger operator containing (normal) Zeeman effect and Coulomb interaction terms will be included. ??2, 3, and 4 will give a brief account of functional integration, semigroup theory and quadratic forms on a Hilbert space, respectively. They will be needed for the main body of the paper consisting of ??5 and 6. ?5 will give the statement of the theorems along with certain remarks. ?6 will contain the proofs of the theorems.

Introduction. The main object of study in this paper is the space of all bounded complex-valued functions defined on a metric space (X, d) which satisfy a Lipschitz condition with respect to the metric d. With a suitable norm this collection of functions becomes a Banach algebra. This Banach algebra, which we shall call a Lipschitz algebra, will be denoted by Lip(X, d). It is the structure of Lip(X, d) with which we are primarily concerned here. Although the notion of Lipschitz continuity is very old and Lipschitz functions have been studied for many years, interest in the Banach space and Banach algebra theory of Lipschitz functions has not developed until quite recently. Very little is known about the Banach space properties of Lip (X, d). The following papers constitute all the work known to this writer which has been done in this area. Mirkil [9] and de Leeuw [4] have considered the space of periodic Lipschitz functions on the real line (or in other words, Lipschitz functions defined on the circle group). Mirkil was interested in the relation between the Lipschitz condition and the translation properties of functions, the fact that the functions were defined on a group being vital. de Leeuw was more directly concerned with the Banach space structure, his paper involving the study of certain dual spaces, extreme points, and isometries. In the long paper of Glaeser [6], a short chapter, somewhat off the main theme of his work, is devoted to the space of n-times continuously differentiable functions defined on a subset of Emn whose nth derivatives satisfy a Lipschitz condition. It is shown as a side result in the paper of Arens and Eells [1] that Lip(X,d) is always the dual space of some normed linear space. The only work known to us which treats Lip(X,d) as a Banach algebra was done by S. B. Myers [11]. His paper [11] is a summary of results, the proofs never published because of his untimely death in 1955. His interest in Lipschitz algebras was connected with the process of differentiation, and this interest we pursue in Chapter III of the present study. We supply proofs of many of the unproved statements of Myers, and in some cases extend his results to more general settings.

In a recent paper R. S. Strichartz has extended and simplified the proofs of a few well-known results about integral operators with positive kernels and singular integral operators. The present paper extends some of his results. An inequality of Kantorovic for integral operators with positive kernel is extended to kernels satisfying two mixed weak LP estimates. The "method of rotation" of Calderon and Zygmund is applied to singular integral operators with Banach space valued kernels. Another short proof of the fractional integration theorem in weighted norms is given. It is proved that certain sufficient conditions on the exponents of the LP spaces and weight functions involved are necessary. It is shown that the integrability conditions on the kernel required for boundedness of singular integral operators in weighted LP spaces can be weakened. Some implications for integral operators in R" of Young's inequality for convolutions on the multiplicative group of positive real numbers are considered. Throughout special attention is given to restricted weak type estimates at the endpoints of the permissible intervals for the exponents. 0. Introduction. The present paper attempts to answer a few questions raised by a paper of Strichartz [25] and the following example may be illustrative of its contents. As usual, for x e RT let lxi=( t= x2)"2. It was shown by Stein [22] that if Q is positively homogeneous of degree zero, bounded and of mean value zero on the unit sphere Snin Rn and K(x)= Q(x)lxl -n then the singular integral operator T defined by Tf(x) =p.v. K *f(x) = lim,,o fIx-ti > K(xt)f(t) dt satisfies {lTf(x)lPlxla'P dx 2(n -1) and conjectured that the condition q> 2(n 1) is not necessary (Theorem 6 of [25]). It is proved in the present paper that 1/q= 1IaI/n is sufficient which in particular shows that q>2(n1) is not needed (Proposition 7). NOTATION. As in [25] let (X, E, 4u), (Y, Z', v) be totally a-finite measure spaces, K(x, y) a measurable function on Xx Y and define Tf(x)= f K(x, y)f(y) dy where Received by the editors January 9, 1970 and, in revised form, May 29, 1970. AMS 1969 subject classifications. Primary 4425, 4428, 4430, 4432, 4450.

This paper contains some basic results about analytic centers and analytic diameters, concepts which arise in Vitushkin’s work on rational approximation. We use Carathéodory’s theorem to calculate β ( K , z ) \beta (K,z) in the case in which K is a continuum in the complex plane. This leads to the result that, if g : Ω ( K ) → Δ ( 0 ; 1 ) g:\Omega (K) \to \Delta (0;1) is the normalized Riemann map, then β ( g , 0 ) / γ ( K ) \beta (g,0)/\gamma (K) is the unique analytic center of K and β ( K ) = γ ( K ) \beta (K) = \gamma (K) . We also give two proofs of the fact that β ( g , 0 ) / γ ( K ) ∈ co ( K ) \beta (g,0)/\gamma (K) \in {\text {co}}\;(K) . We use Bieberbach’s and Pick’s theorems to obtain more information about the geometric location of the analytic center. Finally, we obtain inequalities for β ( E , z ) \beta (E,z) for arbitrary bounded planar sets E.

It is shown that an ergodic measure preserving transformation with quasi-discrete spectrum is conjugate to: (a) the skew-product of an ergodic measure preserving transformation with discrete spectrum and a measurable family of totally ergodic measure preserving transformations with quasi-discrete spectrum; (b) a factor of the direct product of an ergodic measure preserving transformation with discrete spectrum and a totally ergodic measure preserving transformation with quasi-discrete spectrum. Sufficient conditions are given to insure that an ergodic measure preserving transformation with quasi-discrete spectrum is conjugate to the direct product of an ergodic measure preserving transformation with discrete spectrum and a totally ergodic measure preserving transfonnation with quasi-discrete spectrum. 0. Introduction. The structure of ergodic measure preserving transformations with discrete spectrum and totally ergodic measure preserving transformations with quasi-discrete spectrum has been well studied (cf. Halmos [4, pp. 46-50], Abramov [1], and Hahn and Parry [2]). The purpose of this paper is to reduce as much as possible the study of ergodic measure preserving transformations with quasi-discrete spectrum that are not totally ergodic to the above two cases. We shall show (Theorem 4A4) that an ergodic measure preserving transformation T with quasi-discrete spectrum is conjugate to a skew product S x ISXj where S has discrete spectrum and where Sx is totally ergodic with quasi-discrete spectrum and whose quasi-eigenfunctions are independent of x. Next (Theorem 4.5) if the torsion subgroup of the group of eigenvalues of T is the direct product of a complete group and a group of bounded order, then T is conjugate to the direct product S1 x S2 where Si has discrete spectrum and S2 is totally ergodic with discrete spectrum. An example (?5) is given to show that if the torsion subgroup of eigenvalues of T is not the direct product of a complete group and a group of bounded order, then in general a factorization as in Theorem 4.5 is not possible. Finally using the above results it is shown (Theorem 4.6) that an ergodic measure preserving transformation with quasi-discrete spectrum is conjugate to a factor of the direct product of a ergodic transformation with discrete spectrum and a Received by the editors April 23, 1973. AMS (MOS) subject classifications (1970). Primary 28A65; Secondary 22D40.

In the main body of the paper the results anid proofs are giveni for the general case of a rieal funietionl of ni independent variables, and for a domaini of definition of the given function that is very general from the point of view of analysis situs. For the purposes of all introduction one of the priiicipal results will be giveni for the case where n 2. It mutst be stated, however, that the results for the general case differ tremendously from those for the case n 2. The resuilts for the case 2 can conveniently be given in geometric language as follows: Let i conlsist of the interior points of a finite, connected region of the x, y plane, bounded by a finite number of distinct closed curves without multiple points of any sort, possessiing at each point a tangent which turns continuously with movement of its point of contact. Denote the boundary of X by B. Let f(x, y) be a real function of x and y defined and cointinuous in II and oni B, and possessingo continuous third order partial derivatives in R taking oni continuous bounidary values on B. By the positive direction of the normal to B will be understood the direction that leads across B from points in R to points iiot in R. It is assumned that the unilateral, directional derivative of Jf(x, y) on the side of the inner half of the normal and in the positive directioii along the niornal is everywhere positive on B. Let z -f (x, y) be interpreted as a surface in ordiinary euclidean (x, y, z) space. By a critical point of f (X, y) is meant a point in the x, y plane corresponding to which the tangent plane to the surface z f (X, y) is parallel to the x, y plane. In the neighborhood of such a critical point let f (X, y) be expanded according to Taylor's formula ill powers of differences of the x's and y's wliich vanish at the critical point, and with the remainder as a term of the third order. WVe hereby assume that in this expansion the discriminant of the terms of the second order is not zero. From these assumptions it follows that the critical points are isolated and finite in number, anid that they are of the two general types called

Ascoli's t orem deals with continuous functions and states that the space of bounded, equi-continuous functions is compact. The present paper extends it to the measurable functions. Both the space of bounded qui-measurable functions, and its product with the space of bounded equicontinuous func ions re compact. (Author)

in which (i) X2 is a large complex parameter; (ii) s is the real variable over some interval a < s < b; (iii) 02(s) is a real function which has precisely two zeros, both simple, on the interval (a, b); (iv) X(s, X) is bounded in s and X, and is integrable in s. The zeros of 02(s) are referred to as turning points of the differential equation. The solution forms of a differential Equation (1.1) over an interval that includes just one turning point are known, and can be expressed asymptotically with respect to X through the use of Bessel functions. That being so, it could well seem that forms having validity over the entire interval with two turning points might be obtainable by dealing with the interval by halves, and, in the familiar way, patching together the forms appropriate to the respective halves, along with their derivatives. That procedure, however, is not practicable when the forms in hand are not actual representations but only the leading terms of asymptotic ones. For such a form is explicit only to a certain degree, and because of that ordinarily represents not just a single particular solution over an appropriate x-interval, but a whole infinity of them. In other sub-intervals these solutions may be wholly dissimilar. To fit together two such forms, along with their derivatives, at some chosen point between the two turninlg points, therefore does not identify the solutions to which that fit applies, and accordingly cannot be taken as a basis for inferring the continuations of those solutions in intervals beyond the turning points. In the present paper the interval is not treated by halves. It is shown that forms which are valid over the whole interval can be given in terms of ap

considering such an integral as a generalization of a Taylor series. All developments of that paper were on the assumption that f(x) permitted of the integral representation (1.1). We wish to study here conditions on f(x), both necessary and sufficient, for the validity of such representation. Following the analogy of Taylor's series we might at first be tempted to suppose that the analyticity of f(x) in a half-plane, the region of convergence of an integral (1.1), would be the condition required. That this is not the case we see at once by recalling that such a function as sin x, analytic in the entire plane, admits of no representationt of the form (1.1). We are led, however, to a correct conjecture by considering our problem as the analogue of the moment problem of F. Hausdorff . ? This is the problem of determining a function X(x) bounded and non-decreasing in the interval O?? _ and such that