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LET X be a bounded combinatorial m-manifold and let Y be a combinatorial n-ball. The following question sometimes arises. Given a piecewise linear embedding of the boundary 8 of X in the boundary Yof Y, can it be extended to an embedding of X in Y? The main result in this direction derives from Irwin’s Embedding Theorem [2]. Since any map of X into Ymay clearly be extended to a map of X into Y, Irwin’s result in this case may be stated as follows.

Suppose thatbelongs to L2( − π, π). If most of the coefficients vanish then f (x) cannot be too small in a certain interval without being small generally. More precisely Ingham ((2), Theorem 1) has proved the followingTHEOREM A. Suppose that f (x) is given by (1·1) and that an = 0, except for a sequence n = nν, where nν+1 − nν ≥ C. Then given ∈ > 0 there exists a constant A (∈), such that we have for any real x1

1# Introduction. Consider a compact Hausdorff space X and the set C(X) of all continuous complex-valued functions on X, Consider also a subset 21 of C(X) which is an algebra, which is closed in the uniform topology of C(X), which contains the constant functions, and which contains sufficiently many functions to distinguish points of X. Such an algebra 21 is called self-adjoint if the complex conjugate of each function in 2t is in 21. The classical Stone-Weierstrass Theorem states that if 21 is self-ad joint then 21 = C(X). If 21 has the property that the only functions in 21 which are real at every point of X are the constant functions then 21 is called anti-symmetric. Clearly anti­ symmetry and self-adjointness are opposite properties, in the sense that if 21 has both properties then X must consist of a single point. Hoffman and Singer [2] have studied these two properties and given several interesting examples. The present paper was inspired by their work but it more directly relates to a previous paper of Silov [3]. The purpose of the present paper is to prove the following decomposition theorem for a general algebra 21 of the type defined above.

We prove here some polynomial approximation theorems, somewhat related to the Szasz-Mfintz theorem, but where the domain of approximation is the integers, by dualizing a gap theorem of C. l ~ Y I for periodic entire functions. In another Paper [7], we shall prove, by similar means, a completeness theorem ibr some special sets of entire functions. I t is well known (see, for example [l]) tha t i f E is the space of all entire functions in the topology of uniform convergence on compact sets, then the dual space of continuous complex-valued linear functionals on E may be represented as E0, the space of entire functions of exponential type. Now let E (1) b e the space of entire functions of period 1. Then it may be shown that the dual of E (1) can be represented as E0 (l), where E0(1) is the following quotient space of E0: define / ~ g for functions 1, g e E0 i f / _ g is a multiple of sin z~z, and let E0 (l) be the space of equivalence classes of E0 modulo this relation of equivalence. Now E0(1) is apparent ly the same space as the space of restrictions of functions in E0 to the integers Z. Each such restriction is just a two-sided sequence of complex numbers, of at most exponential growth. ConVersely, it is easy to interpolate any such sequence by an entire function of exponential type. Thus, the dual of E(1) is just the space of all such sequences. Actually, we establish this identification by another procedure. To any theorem about periodic entire functions will correspond a theorem about the space of sequences described above. In [6], C. RI~NYI proved an interesting gap theorem, reproduced below. We show by means of duality tha t certain theorems of polynomial approximation are equivalent to this theorem. The domain of approximation is the integers in one case, and the positive integers in another ease. To our knowledge, the problem of polynomial approximation on the integers has not been Considered except in the note [3]. We know of no direct proof of our results. Ultimately, the R ~ Y I result depends on a simple application of Rolle's theorem. I t Would be of interest to have more precise gap theorems than the R~NYr theorem and also to have direct proofs of the results we prove by means of it.

The Darmois-Skitovic Theorem on characterization of normal lew through independence of linear functions of random variables is extended to independence of more general functions satisfying an addition theorem.

0. Introduction. In this paper we study a sequence of classes of computable functions for which a prediction of the complexity of the calculation may be made in a comparatively simple fashion. The class of recursive functions is accepted as being the class of just those functions for which there is a mechanical procedure for obtaining the values from the arguments. However, this class, since it involves all computations, must involve arbitrarily complex computations. Various more restricted classes of functions have been proposed as being somehow more easily computable, but the greater ease of computation has largely been left on the level of intuition. The notable exception to this is the class F of functions computable(') by finite automata. However, this class is too severely restricted; it does not even include multiplication. We present a hierarchy of more inclusive classes of functions, each defined as the class of functions whose computational complexity is "predictable" by a function in the preceding class. It is known that the class F of numerical functions computable by finite auto

Jacques Hadamard died on 17 October 1963 at the age of 98. He published his first mathematical paper of importance in 1888, and continued working until he was over 90 covering an immense range of mathematical subjects including educational, philosophical and psychological aspects of mathematics. To classical analysts his name has been well known as the author of the Hadamard gap theorem, the Hadamard three circle theorem , the Hadamard factorization theorem for integral functions and other results on Taylor series published before 1900, but perhaps it was his proof of the prime number theorem in 1896 more than anything else which made Hardy describe him in 1944 as the ‘living legend’ in mathematics. To some pure mathematicians he may be better known by his theorem on the modulus of a determinant which plays such an essential part in the Fredholm theory of integral equations, or because he invented the name for functional analysis. His work on the theory of propagation of waves and partial differential equations was no less significant, but it is less easy to pinpoint particular results. It is at the base of the modern theory of both subjects; it includes much work on the Cauchy problem and on the technique of the finite parts of integrals which, although superseded by the theory of distributions, proved to be very useful. He was no physicist, but he helped to lay the foundations of the modern theory of shock waves. No one person could do justice to such an enormous range of mathematical activity which was matched by wide interests outside mathematics, and what I have to say owes much to the assistance of others. In particular Mile Jacqueline Hadamard has supplied me with much information and copies of articles and speeches about her father, but, owing to the fact that all their belongings were stolen by the Germans during the war, and also, because as she admitted, ‘he had so little order and method, keeping everything about others and so little about himself’, the information, and in particular the list of published work, is probably incomplete, and the latter is certainly difficult to check. There are over 300 items, many of them published long ago in periodicals not easily traced from the brief descriptions given in the list in the Selecta [284].* I have drawn largely from speeches made at his Scientific Jubilee [284a] in 1936, from articles by Lévy, Mandelbrojt, Fréchet, Julia and Kahane and from Hadamard’s own accounts of events and influences extracted from various articles, books and speeches. I am also particularly grateful to Professor Fréchet, Professor Leray, Professor Kahane, Professor Temple, Professor G. N. Ward, Mr A. E. Ingham, Professor Truesdell, Dr F. G. Friedlander, Dr L. S. Bosanquet and Sir Edward Collingwood for their help. * Numbers in square brackets refer to the numbered items in the bibliography described at the end of the memoir, pp. 96 to 99.

2. G. Higman, On finite groups of exponent five, Proc. Cambridge Philos. Soc. 52 (1956), 381-390. 3. A. I. Kostrikin, On Burnside's problem, Dokl. Akad. Nauk SSSR 119 (1958), 1081-1084. (Russian) 4. M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. tcole Norm. Sup. (3) 71 (1954), 101-190. 5. H. Zassenhaus, Ein Verfahren, jeder endlichen p-Gruppe eine Lie-Ring mit der Charakteristik p zuzuordnen, Abh. Math. Sem. Hamburg Univ. 13 (1939), 200-207.

An example is given of a uniformly continuous constructive function f ( x , y ) f(x,y) with f ( 0 , 0 ) = 0 f(0,0) = 0 , such that the differential equation y ′ = f ( x , y ) y’ = f(x,y) with the initial condition y ( 0 ) = 0 y(0) = 0 has no constructive solution.

This paper continues investigations pertaining to recursive bounds on computing resources (such as time or memory) and the amount by which these bounds must be increased if new computations are to occur within the new bound. The paper proves that no recursive operator can increase every recursive bound enough to reach new computations. In other words, given any general recursive operator F[], there is an arbitrarily large recursive t() such that between bound t() and bound F[t()]() there is a gap in which no new computation runs. This demonstrates that the gap phenomenon first discovered by Borodin for composition is a deeply intrinsic property of computational complexity measures. Moreover, the Operator Gap Theorem proved here is shown to be the strongest possible gap theorem for general recursive operators. The proof involves a priority argument but is sufficiently self-contained that it can easily be read by a wide audience. The paper also discusses interesting connections between the Operator Gap Theorem and McCreight & Meyer''s important result that every complexity class can be named by a function from a measured set.

It is proved that, for any m given distinct real numbers a1, …, am, there exist transcendental entire functions f(z) at most of order m for which all the valuesare rational integers.

This theorem is an improved version of the result obtained by Myhill [1] as well as a similar one obtained by Routledge [2, Theorem 4]. As functions of one variable are concerned, the foregoing theorem is the same as Routledge's except that the well-ordering < referred to in this note is primitive recursive while it is defined by general recursive processes in Routledge's paper (see [2, Theorem 2 ]). Myhill's statement is, however, not sufficiently explicit. A more explicit statement of his result was supplied by Kleene in a letter to the author. The theorem in this note differs from Kleene's formulation only in that Kleene expressed f(n) as U(g(2n)) while here the function U is not used. This leads to the result f(n) =g(2n), which is, according to Routledge (see [2, ??8 and 9]), the best possible in the sense that we can no longer omit the function 2n and therefore can not simply equate f(n) and g(n). PROOF OF THE THEOREM. Let f(n) be any given general recursive function. According to a theorem by Kleene [3, p. 288] there are two primitive recursive functions K(n) and T(x, y) such that

One of the more useful theorems of plane topology was proved virtually simultaneously by Rutt [3] and Roberts [2].2 As modified by Moore [1, p. 296], it is essentially this: Suppose that in the plane or the 2-sphere there exist two points, a and b; a collection, G, of continua whose union is a compact set, M, not containing a or b; and a continuum, C, such that the intersection of each two elements of G is precisely C. Then if no element of G separates a from b, neither does M. This is, of course, a form of addition theorem. Even in 3-space this result is not true as stated, for the collection of circles given in rectangular coordinates by x2+y2+z2-x=O, az=bx, for all a, b, satisfies all the conditions on G with respect to the points (1/2, 0, 0), (2, 0, 0), whereas their union separates these points. There is, however, a theorem concerning linking, which is valid in quite general spaces, and which reduces to the above theorem in the plane.

The well-known Jackson theorem gives estimates on the rate of approximation of functions in C[0,1] by means of polynomials \({\text{P}}_{\text{n}} \left( {\text{x}} \right) = \sum\limits_{{\text{k = o}}}^{\text{n}} {{\text{a}}_{\text{k}} {\text{x}}^{\text{k}} } \) of degree ⩽ n.

These lectures deal with the field of interaction between the theory of analytic functions and functional analysis. In the first section, we shall study the classical theory of analytic functions of a single complex variable, using functional analysis as a tool to obtain some basic classical theorems like Cauchy’s integral theorem and the Mittag-Leffler theorem. In the second section, we dualize a gap theorem of C. Renyi for periodic entire functions to obtain some approximation theorems. One of these is a theorem about weighted polynomial approximation on the integers. In the third section we make a new application of perturbation of a basis in a Banach space to prove uniqueness theorems for analytic functions of one and of several complex variables. In the fourth section, we sketch a theory of duality of vector spaces of entire functions that leads to the solution of the problem of spectral synthesis for mean-periodic entire functions of one complex variable. The remainder of the lectures is devoted to a study of the space of bounded analytic functions in various weak topologies.

This paper continues investigations pertaining to recursive bounds on computing resources (such as time or memory) and the amount by which these bounds must be increased if new computations are to occur within the new bound. The paper proves that no recursive operator can increase every recursive bound enough to reach new computations. In other words, given any general recursive operator F[ ], there is an arbitarily large recursive t( ) such that between bound t( ) and bound F[t( )]( ) there is a gap in which no new computation runs. This demonstrates that the gap phenomenon first discovered by Borodin for composition is a deeply intrinsic property of computational complexity measures. Moreover, the Operator Gap Theorem proved here is shown to be the strongest possible gap theorem for general recursive operators. The proof involves a priority argument but is sufficiently self-contained that it can easily be read by a wide audience. The paper also discusses interesting connections between the Operator Gap Theorem and McCreight & Meyer's important result that every complexity class can be named by a function from a measured set.

The complexity closure of a computable function is defined by a set of axioms. The axioms are satisfied by complexity classes that are computation time closed and also by other complexity classes which do not have this property. It is then shown that there exist honest recursive functions whose complexity closure are setwise incomparable. Further that there exist chains of honest recursive functions whose complexity closures are densely ordered under set inclusion.