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Introduction. In a recent paper, [l], A. Beurling has shown that the positive translates of an integrable function defined on [0, m0) generate, in a certain sense, at least one exponential of the form e-iz, x > O, Ia < 0, provided that the function does not vanish outside a finite interval. It is the converse problem with which we shall be concerned here; namely, to what extent can the exponentials so generated be used to approximate the given function. We are able to give what amounts to a complete solution. The situation resembles strongly that of Schwartz' theory of mean periodic functions [2]. M. Kahane has shown in [3] (see also [41) how this theory can be presented very simply using the notion of Fourier transform of a mean periodic function. Beurling also made use of this method in the present case; however, we shall find it convenient to exploit this tool more systematically, in closer analogy with Kahane's work. We shall also study our approximations in a topology (the same as the one used in the theory of mean periodic functions) which is simpler than that of Beurling. Beurling based his work mainly on a certain division theorem which states roughly that an entire function is of finite exponential type if it is bounded on a half plane and equal to the ratio of two bounded analytic functions on the complementary half plane. The conclusions we make here follow from a refinement of this given in ?3 which yields an upper bound for the type of such an entire function. It should be remarked that B. Nyman ([7, pp. 28-29]) has established a result similar to the one given here, using, however, a quite different topology. (The referee calls attention to this in his report; although I have since had the opportunity to consult Nyman's work, it was not accessible to me in New York at the first writing of this paper.)

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

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

By using the results of [6], it is proved that for an extensive class of increasing functions h, ( ∗ ) ∑ 1 ≦ d ≦ x μ ( d ) d h ( x d ) ∼ x h ′ ( x ) as x → ∞ \begin{equation}\tag {$*$}\sum \limits _{1 \leqq d \leqq x} {\frac {{\mu (d)}}{d}h\left ( {\frac {x}{d}} \right )} \sim xh’(x)\quad {\text {as}}\;x \to \infty \end{equation} where μ \mu denotes the Möbius function. This result incidentally settles affirmatively Remark (iii) of [6], and refines the Tauberian Theorem 2 of that paper. It is also shown that one type of condition imposed in [6] is necessary to the truth of the cited Theorem 2, at least if some sort of quasi-Riemann hypothesis is true. Nevertheless, examples are given to show that on the one hand ( ∗ ) ( ^\ast ) may be true for functions not covered by the first theorem of this paper, and on the other that some sort of nonnaïve condition on a function h is necessary to ensure the truth of ( ∗ ) (^\ast ) .

2. Preliminary definitions and theorems. Throughout this paper all integrals considered will be Hellinger [1i type limits of the appropriate sums (the definitions, theorems and proofs of this paper can be extended to "many valued" functions). Thus, if K is a real valued function of subintervals of the number interval [a, b], the existence of f[a,bjK(I) is necessary and sufficient for the existence of f1K(J) for each subinterval I of [a, b]. Furthermore, the function F of subintervals of [a, b], defined by F(I) =fK(J), is additive. If K is a real valued function of subintervals of [a, b], then the statement that K is 1-bounded on [a, b] means that there is a subdivision D of [a, b] such that the set of sums EEK(I), where E is a refinement of D and the sum is taken over all intervals I of E, is bounded. This implies that if I is an interval in a refinement of D, then the set of sums EQ K(J), where Q is a subdivision of I and the sum is taken over all intervals J of Q, has a least upper bound L(I) and a greatest lower bound G(I). We now see that if each of R and T is a refinement of D, and S is a refinement of each of R and T, then ER G(I)-< Es G(I) -< s K(I) ? Es L(I) ? E2T L(I). From this it follows that each of f(a,b]G(I) and f[a,b]L(I) exists, that f[a,b]G(I) < f[a,b]L(I), and that f(a,b]K(I) exists if and only if f[a,b]G(J) f[a,b]L(I). We state without proof a theorem of Kolmogoroff [2].

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 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

The objective of this paper is to give an alternative derivation of results on bounded analytic functions recently obtained by Ahlfors [1] and Garabedian [2].1 While it is admitted that the main idea to be used is more in the nature of a lucky guess than of a method, it will be found that the gain in brevity and simplicity of the argument is considerable. As a by-product, we shall also obtain a number of hitherto unknown identities between various domain functions. The basic problem treated in the above-mentioned papers is the following generalization of the classical Schwarz lemma: Given a finite schlicht domain D of connectivity n (n> 1) in the complex zplane and a point r in D, to find a function F(z) with the following properties: (a) F(z) belongs to the family B of analytic functions f(z) which are single-valued and regular in D and satisfy there If(z) I _ 1; (b) I F'(r) I > lf'(r) J, where f(z) is any function in B. Evidently, it is sufficient to solve this problem for any domain D' which is conformally equivalent to D. In particular, we may therefore assume, without restricting the generality of what follows, that D is bounded by analytic curves. It was shown by Ahlfors that F(z) yields a (1, n) conformal mapping of D onto the interior of the unit circle and that n-I of the n zeros of F(z) coincide with the zeros of a single-valued function h(z) which is regular in D with the exception of a simple pole at z = and satisfies -'ih(z)dz>O on the boundary r of D; the nth zero of F(z) is located at zx=. It was subsequently noticed by Garabedian that the function h(z) can be written in the form h(z) = F(z)q(z) where q(z) (Z )-2 is regular in D and that the extremal property of F(z) can be deduced in a very elegant manner from the resulting inequality

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

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

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

1. Some years ago Collingwood and Valiront proposed the problem of whether there could exist an integral function whose minimum modulus on every circle Iz I =r is bounded, but possessing no asymptotic paths. By an asymptotic path we mean a continuous path tending to infinity along which the value of the function tends to a limit. In this paper I show how to construct such a function. It is obtained by considering the well known Weierstrassian non-differentiable function

The objective of this paper is to present an estimate bounding the L 2 {L_2} -norm of a function over a bounded subset of R n {R^n} by the L 2 {L_2} -norms of its derivatives of arbitrary order over all of R n {R^n} and the L 2 {L_2} -norm of its projection onto a finite-dimensional space of functions with bounded support. The estimate essentially generalizes inequalities of Friedrichs [1, p. 284] and Lax and Phillips [2, p. 95]. An application of the estimate is made to the Fredholm theory of elliptic partial differential operators in R n {R^n} .

1. Introduction. In this paper we prove, by means of Hadamard's variational formula, several monotonicity theorems for the electrostatic capacity of a solid or a lamina. From these monotonicity theorems follow some well known isoperimetric inequalities for the capacity which are, however, not such strong statements as the monotonicity theorems themselves. A point of interest in the development as we present it here is that, up to this time, it has rarely been found possible to solve extremal problems and obtain isoperimetric inequalities directly using only Hadamard's formula, since the validity of the formula depends on piece-wise smoothness of the boundary of the domain under consideration. 2. Volume and capacity. Let D be a finite 3-dimensional domain including the origin in its interior and bounded by a piece-wise smooth surface S. We denote by G =-t + 0(r) r the Green's function of D for Laplace's equation, with a pole at the origin, where r is the distance from the origin. We shall call 7 the inverse capacity of D with respect to the origin, and we shall denote by V the volume of D. Obviously, 7-1 has the dimension of length. Let a deformation of .S be described by an infinitesimal shift of 5 through 5w units along its inner normal n; we obtain the well known Hadamard variational formulas [2]

J. E. McMillan [6] has shown that if ? is an open disk in the plane and if f is continuous in ?, then the set of curvilinear convergence of f is of type Fa. In this paper we prove that there exists a bounded continuous complex-valued function f, defined in the interior of a three-dimensional cube, such that the set of curvilinear convergence of f is not a Borel set. Thus McMillan's theorem does not generalize to three dimensions. However, the following question remains open.

1. Object and results. In this paper I define for every function f (x), bounded and summablet on an interval a x b, or a measurable subset thereof, two numbers, the measurable upper and lower bounds of f (x) on (a, b). These numbers are analogous to the extrema extremorum of a function which is continuous on (a, b), with which in fact they coincide when f (x) is continuous. I show that the measurable bounds can be determined by means of an enumerable set of constants (the " momental co1stants ") defined by the Lebesgue integrals

have been determined.t It is easy to construct sequences for which the limit on the left side of (1) exists and is different from the right side. In the present paper necessary and sufficient conditions are obtained for the existence of this limit in terms of the sequence s,(x). It turns out that the limit function F(x) is independent of f(x). In fact if F(x) is an arbitrary continuous function, there exists a sequence of summable functions sn(x) tending to zero everywhere, for which

where y *(x) =min (yn(x), 1). Under the additional assumption that G(y) satisfies a Lipschitz condition on [0, 1 ] they proved that the sequence yo(t), yi(t), * * * converges to a bounded solution,' y(t), of (1). Dr. Mann pointed out to me that it was not known whether or not the requirement of a Lipschitz condition was superfluous. The present paper resolves this uncertainty by giving an example of a function G(y) satisfying (2) for which the corresponding sequence (3) does not converge. It also contains a positive result, to the effect that the sequence defined by (3) does converge to the solution y(t) if, in addition to requirement (2), G(y) is convex.

shall be a minlimum, where f(x) is a given continuous ftinction, and e (x) is a given bounded function with a positive lower bound. The purpose of the present paper is to discuss the somewhat less simple problem of convergence that arises if the greatest lower bound of e (x) is zero. There will be occasion incidentally to develop certain other topics, which have some independent interest: in ?? 2 and 3, the theory of the existence and uniqueness of discontinuous approximating functions of a high degree of generality, and in ? 5, some corollaries of Bernstein's theorem on the derivative of a trigonometric sum. 2. General theorem on existence of an approximating function. Let

1. Introduction. This paper presents some properties of the functions in a certain subclass of the class of subharmonic functions. Consider real valued functions v(z) defined in a domain D of the complex plane. DEFINITION 1. The function v is in (hs), the class of functions which have harmonic support in D, if and only if v is uniformly bounded on each compact subset of D and for each zoC=Dah harmonic and single valued in D3h(zo) =v(zo) and h

A blocking set of a function j is a closed set which does not intersect f but which intersects each continuous function with domain the same asf. It is shown that for each function which is not almost continuous, there exists a minimal blocking set. Using this property it is shown that there exists an almost continuous function with domain [0, 1] which is a G5 set but is not of Baire Class 1, and that there exists an almost continuous function dense in the unit square. Unless otherwise stated, all functions considered are real functions with domain a closed and bounded subset of the real line, R. No distinction is made between a function and its graph. If the functionf is a connected subset of the plane, f is simply said to be a connected function. If each open set containingf also contains a continuous function with the same domain as f, then the function f is said to be almost continuous. Stallings [7] showed that if the functionf is almost continuous and has connected domain, then f is connected. He stated as an open question the following. Is each connected function almost continuous? This question was answered in the negative by several authors ([1], [2], [4], [6]). In [4], Jones and Thomas showed that there exists a function which is not of Baire Class 1, is a G5 set, is connected but is not almost continuous. In Example 1 of the present paper, it is shown that there exists such a function which is almost continuous. Example 2 shows that there exists an almost continuous function which is a dense subset of the unit square. In [1], Brown gave an example of a connected but not almost continuous function dense in [0, 1] x R. Suppose M is a subset of the plane. The X-projection of M is denoted by (M)x. The Y-projection of M is denoted by (M)y. If Kc (M)x, MK denotes the part of M with X-projection K. The function f is said to be of Baire Class 1 if f is the pointwise limit of a sequence of continuous functions. Suppose that D is a set containing the functionf The statement that the subset C of D is a blocking set off relative to D means that C is closed Received by the editors June 15, 1971. AMS 1970 subject classifications. Primary 54C10, 54C30, 26A15; Secondary 26A21.

This theorem represents the best result that can be established for uniform polynomial approximation to functions which are merely continuous; the converse may be found in J. L. Walsh's work on approximation [5]. The aim of this paper is to extend Lavrentieff's theorem by establishing it on a certain class of sets which do separate the plane. The methods used will be similar to those first introduced by M. Keldysh [2 ] and S. Mergelyan [3 ]. That the possibility of uniform approximation to continuous functions cannot be established for all closed bounded sets with no interior may be seen from the following counterexample, presented by V. A. Tonyan [4]: Let E be the closed set with no interior obtained by removing from the unit disk a countable number of open disks k, which are distributed everywhere densely in the disk, and whose circumferences r, are tangent neither to each other nor to the unit circumference. Let us further choose these disks in such a way that their radii csatisfy E=1 e^ 0 is a given fixed number. Now suppose that for any function f(z) continuous on E there exists a rational function R(z) such that

Existence theorems are proved for basic problems of Lagrange in the calculus of variations and optimal control theory, in particular problems for arcs with both endpoints fixed. Emphasis is placed on deriving continuity and growth properties of the minimum value of the integral as a function of the endpoints of the arc and the interval of integration. Control regions are not required to be bounded. Some results are also obtained for problems of Bolza. Conjugate convex functions and duality are used extensively in the development, but the problems themselves are not assumed to be especially "convex". Constraints are incorporated by the device of allowing the Lagrangian function to be extended-real-valued. This necessitates a new approach to the question of what technical conditions of regularity should be imposed that will not only work, but will also be flexible and general enough to meet thediverse applications. One of the underlying purposes of the paper is to present an answer to this question. 1. Statement of main results. Let [a, b] be a real interval, and let L be a function on [a, b] x R' x R' with values in (no, + no]. For each subinterval [t0, t1l C [a, b] and endpoint pair (c0, c1) E R' x R', we consider the problem of Lagrange in which the integral

Introduction. There are well-known procedures due to Douglas [6], [7] for finding an approximation to an analytic or harmonic function in the unit disk when data are given at a finite number of points inside. The success of the error analysis depends on estimates which show that an analytic or harmonic function defined and bounded on the unit disk depends continuously on its restriction to a concentric disk of smaller radius. In general, the problem of determining a function in a given class from its restriction is called the continuation problem. When the function exists, is unique and depends continuously on its restriction, the continuation problem is said to be wellposed in the sense of Hadamard. For practical purposes continuous dependence must be taken to mean Holder continuous dependence. When there is Holder continuous dependence, the continuation problem, in the terminology of F. John, is said to be well behaved. The estimates used in the error analysis of Douglas's numerical procedures, due to Hadamard [12] for analytic functions and to Miller [16] for harmonic functions, are an example of this kind. They show that the continuation problem is well behaved for bounded analytic functions and for bounded harmonic functions in the plane. A more general result is due to F. John, who has shown in [14] that the continuation problem for solutions of linear analytic elliptic equations of arbitrary order in any number of variables is well behaved if a bound on the solutions is prescribed. The subject of this paper is the nature of the continuation problem for the solutions of nonanalytic elliptic equations in more than two variables. The main result is that the continuation problem is well behaved for bounded solutions of (a class of) elliptic equations with C1 coefficients and of arbitrary order. Continuous dependence of the solution on its restriction is expressed in terms of the uniform norm.

Let S be a countable topological space. If S is indexed by the positive integers, then real functions on S may be considered as sequences and are subject to infinite matrix transformation. After Henriksen and Isbell in [3], a point s in S is called a heavy point if there exists a regular matrix which sums every bounded real-valued function on S continuous at s. It is clear that the notion of a heavy point is independent of the order chosen for S. Some properties of heavy points are given in [3 ]. For example, sequential limit points are heavy points and isolated points (in fact, points whose complements are C*embedded) are not heavy points. In addition Henriksen and Isbell observe that not all heavy points are sequential limit points. The latter phenomenon was the motivation for the present paper. Corresponding to each positive regular matrix A we define a topology on the positive integers by taking neighborhoods of 1 to be sets whose characteristic functions are A-summable to 1. Denote the resulting space by NA. The point 1 is a heavy point in NA. Some basic properties of these spaces are examined. Also, we determine the class of matrices A for which 1 is not a sequential limit point in NA. Let A = (ank) be an infinite real matrix. The A-transform of the real sequence x = { xk } is the sequence { Sk ankXk } if these sums exist. Let c and m denote the Banach spaces of convergent sequences and bounded sequences, respectively, with jjx|I =supk| X . Let CA = {x: AxEc}. The matrix A is said to sum the members of CA or any subset of CA. Define the functional limA on CA by limAx=lim Ax. The matrix A is called conservative if CCCA and regular if in addition lim =limA on c. It is well known that A is regular if and only if limn ank=O for all k, limn Ekank=l1 and SUpn Zk lank! < o. The matrix A is called positive if ank _ 0 for all n, k. For a set T of positive integers let x(T) denote the characteristic function of T; (X(T))k =1 for k E T, 0 otherwise. For a topological space S let C*(S) denote the Banach space of bounded real-valued continuous functions on S with ||x|| =sup { I x(s)|: sES}. As observed in [3], if S is countable and completely regular, then S is 0-dimensional (i.e. the ring r of open and

The starting point of this paper is the following theorem of Orey [1]: Let T be a linear operator on L1 of the integers (i.e., T acts on absolutely convergent series). Assume (a) T is positive (i.e., if f is a nonnegative function on the integers then so is Tf), (b) ||T||1 O,f Oimplies Tif= oo everywhere, (d) for each pair of integers i, j there is an no(i, j) such that if bi is the function that is 1 on i and 0 elsewhere then Tna8(j) > 0 for all n>nO. Under these conditions if f is a function such that ffdu=O (and f IfI du< oo) then limn, jf Tnfl du =O. Sucheston began the study of operators where condition (b) is replaced by (b') supn||Tn|I1< oo. He shows, using Banach limits, that either limn,wf| TnfI du =0 for all f in L1 or we can find a bounded function e, strictly greater than 0, such that T*e=e. (T* is the adjoint of T.) (See Theorem 1, 2 in part 1 of [2]). He conjectured in a letter to the author that if f|fadu< oo and jffedu=0 then limn,f fI TnfI du =0. The purpose of this note is to prove that conjecture. Let v be the measure induced on the integers by e. (ffdv = ff f edu.) Then ffdv =fTfdv and hence Orey's theorem shows that if ffdv =0 then limnOO0 f Tnf| dv =0. What needs to be shown is that limn,o fI TnfI du = 0. (e may not be bounded away from 0.)

surface 3x=h-'(X) and D(u; X) for the Dirichlet integral over the region ix bounded by a and fi. The main result of this paper is the inequality: maxhJ|Qx=m(h; X)=D(h; X)

A multiplicative function f on G is a complex-valued function on G which satisfies the relation f(uv) =f(u) f(v) for each pair of elements u, v in G. A semi-character on G is a bounded multiplicative function on G that is not identically zero. The set of semi-characters on the commutative semi-group G will be denoted by 0. Many of the results of the present paper depend on the joint work of Hewitt and Zuckerman. They show in [3 ] that the set 1Z of nontrivial multiplicative linear functionals on 11(G), known to be in (1-1) correspondence with the space of regular maximal ideals in 11(G), may be identified with G. Indeed, Theorem 2.7 of [3] shows that if with the function X G 0 we associate the function 7(X) = -rTM defined by r(a) = FEzc a(z)x(z), then v is a (1-1) mapping of G onto M. We give the set G the topology it inherits when considered as a subset of the dual space l1*(G) of 11(G). We recognize this relative topology on 6 as the product, Gel'fand, or w* topology. For each X G G, the set U= 1{iGOI | 61(z) -x(z) I O and F is a finite subset of G, is an open neighborhood of X, and the set 63(X) of such neighborhoods is a fundamental neighborhood system at X. If 0 is isolated from G in 11*(G), then G is compact [3, Theorem 8.5]. In any case G is locally compact, and

Given a sequence of real numbers {aj} with fractional parts , let Nj(a, b) = #{m I 3> 1. So, it is known that {n cos ny} is u.d. (mod 1) for almost all y, and it is easy to show that {cos ny} is not. The question arises: how rapidly mustf increase in order to insure that {f(n) cos ny} is u.d. (mod 1) for almost all y? It is the purpose of this paper to show that any sufficiently smoothf which increases to infinity, no matter how slowly, will serve. In what follows only the following facts about the cosine are used: it is continuous, bounded, twice differentiable, periodic, piecewise monotone, and at no point do both its first and second derivatives vanish. The results below will hold for any function with these properties in place of the cosine. We will suppose throughout that f increases without bound, f' decreases, and f(x) = o(xll2 E) as x -soo for some e > 0. Also, we assume that y/2i7 is irrational. LEMMA 1. Suppose that {Xk} and {Yk} are such that limk.o (Xk-Yk)=0. If either sequence is u.d. (mod 1), so is the other.

In this paper we construct an outer function in HP for all p which is not normal. The purpose of this note is to answer a question raised by Professor Joseph Cima; namely, we show the existence of nonnormal outer functions which are in HP. See [1] for relevant definitions. The existence of a function in HP for all p which is not normal easily follows from a theorem of P. Lappan [2, p. 190] which says that given an unbounded functionf, holomorphic on the unit disc D, then there exists a Blaschke product B such that Bf is not normal. However Bf is clearly not outer. We shall construct an outer function with two different asymptotic values at z= 1, which will imply by a theorem of 0. Lehto and K. I. Virtanen that this function is not normal. THEOREM [3, THEOREM 2, p. 53]. Letf be meromorphic and normal in G, and letf have an asymptotic value a at a boundarypointPalong a Jordan curve lying in the closure of G. Then f possesses the angular limit a at the point P. We present the EXAMPLE. f(z) = oo z exp 2 eit e log t dt} = FF2 ! Log~~ ~~!exP~~2~ itFF. PROOF. f E HP for all p since F2 is bounded and F1 is in HP for all p. From the representation theorem of outer functions, we have that F2 is outer. Furthermore, F1 is a schlicht function and by Theorem 3.17 in [1, p. 51] the singular part of F1 is identically one. Thus F1/z is an outer function and hence f is an outer function. Finally, it is an elementary calculation to show that lim If(e"6)I = 0 and lim If(e"6)I = oo. 0-0+ 0-2ur Received by the editors March 8, 1971. AMS 1969 subject classifications. Primary 3030, 3067; Secondary 3062.