Showing papers in "Transactions of the American Mathematical Society in 1950"

Theory of Reproducing Kernels.

Abstract: : The present paper may be considered as a sequel to our previous paper in the Proceedings of the Cambridge Philosophical Society, Theorie generale de noyaux reproduisants-Premiere partie (vol 39 (1944)) which was written in 1942-1943 In the introduction to this paper we outlined the plan of papers which were to follow In the meantime, however, the general theory has been developed in many directions, and our original plans have had to be changed Due to wartime conditions we were not able, at the time of writing the first paper, to take into account all the earlier investigations which, although sometimes of quite a different character, were, nevertheless, related to our subject Our investigation is concerned with kernels of a special type which have been used under different names and in different ways in many domains of mathematical research We shall therefore begin our present paper with a short historical introduction in which we shall attempt to indicate the different manners in which these kernels have been used by various investigators, and to clarify the terminology We shall also discuss the more important trends of the application of these kernels without attempting, however, a complete bibliography of the subject matter (KAR) P 2

Jordan homomorphisms of rings

Abstract: The primary aim of this paper is to study mappings J of rings that are additive and that satisfy the conditions $$ {\left( {{a^2}} \right)^J} = {\left( {{a^J}} \right)^2},\;{\left( {aba} \right)^J} = {a^J}{b^J}{a^J} $$ (1) Such mappings will be called Jordan homomorphisms. If the additive groups admit the operator 1/2 in the sense that 2x = a has a unique solution (1/2)a for every a, then conditions (1) are equivalent to the simpler condition $$ {\left( {ab} \right)^J} + {\left( {ba} \right)^J} = {a^J}{b^J} + {b^J}{a^J} $$ (2) Mappings satisfying (2) were first considered by Ancochea [1], [2](1). The modification to (1) is essentially due to Kaplansky [13]. Its purpose is to obviate the necessity of imposing any restriction on the additive groups of the rings under consideration.

A theorem of the Hahn-Banach type for linear transformations

Abstract: Every continuous linear functional defined on a vector subspace of a real normed space can be extended to the whole space so as to remain linear and continuous, and with the same norm(2). The extension of continuous linear transformations between two real normed spaces has been studied by several authors and for a long time it has been recognized that this problem has a close connection with the question of the existence of projections of norm one, and moreover that the nature of the space where the transformations take their values is much more important than that of the space where the transformations have to be defined. It is not known, as far as we can say, what are the precise conditions for the possibility of extending a transformation without disturbing its linearity, continuity, and norm. In this paper we shall give a necessary and sufficient condition, which refers only to the space where the transformation takes its values and does not involve the transformation itself or the space where it is to be defined, for such an extension to be possible: the condition is expressed in terms of a certain "binary intersection property" of the collection of spheres of the normed space (see Theorem 1). After that we proceed to the study of the structure of real normed spaces whose collections of spheres have this property. A first step in this direction is given by the theorem asserting that these normed spaces (provided they contain at least one extreme point in the unity sphere) are simply those that can be made into complete vector lattices with order unity in such a manner that the norm derived from the order relation and the order unity in the natural way is identical to the given norm (see Theorem 2, and Theorem 3 for the finite-dimensional case). In this connection we point out a conjecture which we have not been able to settle, namely that, if the collection of spheres of a normed space has the binary intersection property, then its unity sphere must contain an extreme point. By using some results of S. Kakutani and M. H. Stone, we establish the connection between the normed spaces having the binary intersection property and the spaces of real continuous functions over certain compact Hausdorff spaces (see Theorem 4), or the complete Boolean algebras (see Theorem 6).

The hyperplane sections of normal varieties

Abstract: covered in that investigation show, however, that we have in them a class of varieties demanding study for its own sake. Another very good, but in one sense probably more transient, reason for the study of normal varieties is that as yet we are not assured of the existence of a model free from singularities for any given field of algebraic functions, and in fact a greater knowledge of normal varieties may be a prerequisite for the resolution of singularities of arbitrary varieties. Below we direct attention to the question whether, or to what extent, the hyperplane sections of a normal variety(2) are themselves normal. Quite generally, if P is a property of irreducible varieties, we may ask whether the hyperplane sections of a variety with property P share this property. In particular, we may raise this question for the property P of being irreducible. For curves, it is clear that the hyperplane sections will for the most part be reducible, so we shall confine the question to varieties of dimension r ?2. For varieties of dimension r> 2, it is still clear that not all the hyperplane sections will, in general, be irreducible: for example, consider a (suitable) cone; the hyperplane sections through the vertex will be reducible. This example leads us to reformulate the question. The hyperplanes of a projective space in themselves form a projective space, the dual space Sn': we shall say that almost all hyperplanes have the property P, if the hyperplanes not having the property P lie on (though they need not fill out) a proper algebraic subvariety of Sn'. Even if now it turned out to be false that almost all hyperplane sections of an irreducible variety are themselves irreducible, we would not consider the original question on normal varieties as closed, but would reformulate it in local terms; it tturns out, however, that they are irreducible almost always (Theorem 12), and therefore it is possible to deal with the

Means for the bounded functions and ergodicity of the bounded representations of semi-groups

Abstract: 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.

Uniform asymptotic formulae for functions with transition points

Abstract: in which R(x, w), Q(x, w) are regular at x=O, w=0 and f(x) has a simple zero at x=0; for (see ?2) this may be reduced to the form (1.1) by a change of variables, regular at x = z = 0. When all the symbols denote real numbers, the solutions of (1.2) are monotonic or oscillating according as OffQ +4-1R2+ 2-1dR/dx is positive or negative; and when v is large, this quantity changes sign at a point near x = 0, on account of the simple zero of f(x) at x = 0. Thus as x passes through 0 the solutions change from monotonic to oscillating, and we may call x = 0 a transition point('). It is this transition which distinguishes our problem from the simpler one in which the x-region includes no zero of f(x). The guiding idea of the investigation is familiar: "approximately identical differential equations have approximately identical solutions." A significant approximation to (1.1) will be an equation having the same features when z is near 0. Accordingly the project stated in the first paragraph is crystallized: the approximations to solutions of (1.1) are to be solutions of a differential equation of the sameform as (1.1). The simplest equation of this form is the Airy equation

The normal completion of the lattice of continuous functions

Abstract: Let S be a topological space(1) and let C(S) denote the set of all real valued, bounded, continuous functions on S. It is well known that C(S) is a distributive lattice under the operations sup (f,g) and inf (f,g). In general, however, C(S) is not a complete lattice; that is, an arbitrary bounded set of continuous functions in C(S) need not have a least upper bound in the lattice C(S). Furthermore, the structure of the minimal completion of C(S) by means of normal subsets has not been determined even in the simple case where S is the real interval [0, 1].

Abelian group algebras of finite order

Abstract: Introduction. A group G of finite order n and a field F determine in well known fashion an algebra GF of order n over F called the group algebra of G over F. One fundamental problem(') is that of determining all groups H such that HF is isomorphic to GF. It is convenient to recast this problem somewhat: If groups G and H of order n are given, find all fields F such that GF is isomorphic to HF (notationally: GF'HF). We present a complete solution of this problem for the case in which G (and thus necessarily H) is abelian and F has characteristic infinity or a prime not dividing n. The result, briefly, is that F shall contain a certain subfield which is determined by the invariants of G and H and the characteristic of F. 1. Multiplicities. If G is abelian of order n and F is a field whose characteristic does not divide n, the group algebra GF has the structure

On the Schwarz-Christoffel transformation and $p$-valent functions

Abstract: maps the open unit circle |z I < 1 (hereafter denoted by E) onto P the interior of an m-sided convex polygon. The vertices of the polygon are wj =fi(%) and the exterior angle(2) at the vertex wj is yj7r. Conversely if P is given, then z1, Z2 * * * Zm, c1, and c2 can be determined such that (1.1) maps E onto P, and moreover the origin can be carried into any preassigned point of P and the value of argf'(0) can be arbitrarily preassigned. The equation (1.1) subject to the conditions (1.2) and (1.3) is one form of the Schwarz-Christoffel transformation (3). Schwarz(4) stated that the formula (1.1) is easily generalized to the case where P is a multi-sheeted domain bounded by straight lines and containing branch points, and Christoffel(5) considered this generalization in some detail. Study('), Loewner(7), Gronwall(8), Bieberbach(9), Paatero(10), and

ON n-PARAMETER FAMILIES OF FUNCTIONS AND ASSOCIATED CONVEX FUNCTIONS

Abstract: Introduction. Let f(x) be a real-valued function continuous on the interval a < x ? b. Then f(x) is said to be strictly convex if and only if the graph of any linear function for a < x < b meets the graph of f(x) in at most two points. In this situation, one may consider the linear functions on a < x < b as a twoparameter family-for each pair of points (xi, yi) and (x2, y2), xl 5x2, there is exactly one linear graph through these points-and the strictly convex functions as "associated" with the linear functions. Beckenbach and Bing [1, 3](1) generalized this situation by replacing the linear functions by a more general 2-parameter family, that is, a family of continuous functions such that for each pair of points (xi, yi) and (x2, y2), X1i X2, there is one and only one member of the family through these points; then in a natural way they have introduced the associated convex functions. These authors have shown that many properties of the class of linear functions and convex functions hold for 2-parameter families and their associated convex functions. One surprising result was the observation that a 2-parameter family need not be topologically equivalent to the family of linear functions on the interval 0< x <1. T. Popoviciu [9] has given the definition for n-parameter families, but stated no properties. We obtain results here for such families of functions and their associated convex functions which are in part generalizations of those obtained by Beckenbach and Bing. We also obtain results related to the work of T. Popoviciu [7, 8] on convex functions associated with linear families, to that of M. M. Peixoto [6] on the derivatives of generalized convex functions, and to that on approximation discussed by S. Bernstein [4] and C. J. de la Vallee Poussin [5 ]. 1. Definitions and elementary properties. DEFINITION 1. An n-parameter family is a set of single-valued, real, continuous functions f(x) on an interval a < x < b such that for every set of points (xi, yi) (i = 1, * , n) with a ?

On the behaviour of harmonic functions at the boundary

Abstract: 1. Let F(x, y) be a function harmonic for y >0, and suppose that for every point (x, 0) of a set E of positive measure of the x-axis there exist a triangular region with vertex at that point, where the function is bounded; then it is well known that: (A) Almost everywhere in E, F(x, y) has a limit as (x, y) tends nontangentially to (x, 0) G£. This result, first proved by Priwaloff [l](2), when applied to analytic functions leads, as shown by Plessner [2], to the following stronger result: (B) Let F(z), z = x+iy, be a function analytic for y>0. Then, except for a set of points of measure zero, at every point (x, 0) of the x-axis, either the function has a finite limit as z tends nontangentially to (x, 0), or the range of F(z) in every triangular region with vertex at that point is dense in the whole complex plane. Actually these results were proved not only for functions harmonic or analytic in a half-plane, but also in domains limited by rectifiable curves. However, even the special cases mentioned above were obtained by methods of conformai mapping [l], [2] which cannot be applied to harmonic or analytic functions of more variables. A purpose of the present paper is to give a different proof of (A) which leads to its generalization to functions of any number of variables: (a) Let F(P), P = (xx, x2, • • • , x„), be a function harmonic for xn>0 such that for every point Q of a set E of positive measure of the hyperplane x„ = 0 there exists a region Yq limited by a cone with vertex at Q and a hyperplane xn = const, where F(P) is bounded. Then almost everywhere in E the function has a limit as P tends to QÇLE nontangentially to xn = 0. A further generalization of (A), which will enable us to extend (B) to functions of several complex variables, deals with functions which are harmonic in sets of variables, and may be stated as follows: (b) Let E = ExXE2X • • -X-Et» be the Cartesian product of the spaces Ek of points Pk = (x?\\ xf, ■ ■■ , xf ), and F(P), P=(Px, P», • • • , Pm)EE, be defined and continuous in asJfX), k = 1, 2, • • • , m, and harmonic in Pk, that is, such that

Extensions of semigroups

Abstract: By the term semigroup we shall mean a system consisting of a class z of elements, a, b, c, in which there is defined an associative binary operation: a(bc) (ab)c. An ideal of I is a subset S of I such that if a is in 2 and b is in S, then both products ab and ba are in S. Rees(') defines the difference semigroup T = S to be essentially that obtained by collapsing S into a single zero element 0, while the remaining elements of z retain their identity. Thus the T-product of two nonzero elements is defined to be 0 if their I-product lies in S, and otherwise to be the same as defined in 1(2). As in the Schreier theory of group extensions, let us consider the problem of constructing, for given semigroup S and given semigroup T with zero, every possible semigroup z containing S as an ideal, such that z -S is isomorphic with T. Such a z will be called an extension of S by T. If S has a two-sided identity element, every extension of S by T can be obtained by means of a "ramified homomorphism" of T into S (Theorem 2). If S satisfies the mild "Condition A," stated in ?1, every extension of S by T can be obtained by means of a pair of "linked" ramified homomorphisms of T into the semigroups of left and right translations of S (Theorem 3). Condition A is satisfied if S is what Rees (loc. cit. p. 393) calls a completely simple semigroup without zero. This case is of interest because S is then the "Suschkewitsch kernel" of I, originally described by Suschkewitsch(3) for finite semigroups, shown by Rees (loc. cit. p. 392) to be the intersection of all the ideals of 1, and further studied by Schwarz(4) and the author(6). In a previous paper(6) the author gave an extension theorem for such an S by a semigroup T having no divisors of zero. Theorem 5 below extends this to arbitrary T.

On a theorem of Marcinkiewicz and Zygmund

Abstract: The purpose of the present paper is to prove the following result: Let F(P), P= (xi, x2, • ■ • , x„), be a function harmonic for xn >0 and such that for every point Q of a set £ of positive measure on the hyperplane x„ = 0 there exists a region contained in x„>0 limited by a cone with vertex at Q and a hyperplane x„ = const, where the function is bounded. Then except for a set of measure Zero, the integral

Stokes multipliers for asymptotic solutions of a certain differential equation

Abstract: Series (2) converge for I x| < oo and are useful for computing the yj's when x is small. When x is large, convergence is slow and it is more advantageous to use asymptotic expansions. If we let x = rei0 and divide the x-plane into 2p sectors by the radial lines 6=7rh/p; h=O, 1, * , 2p-1, then, as Trjitzinsky [8](1) has shown, there is associated with each of these sectors n independent solutions y,(x) of (1) such that y,(x)-A,(x) where the asymptotic forms

Branch points of solutions of equations in Banach space

Abstract: local solutions of an abstract functional equation was defined and studied. The theory developed was used to obtain existence theorems for integral equations and elliptic differential equations. The purpose of this paper is to extend the multiplicity theory and apply the new results to obtain more existence theorems for integral and differential equations. We study the equation in a Banach space X,

Automorphisms of simple algebras

Abstract: simple (finite-dimensional) algebras. Since the meaning of the designation "Galois theory" has begun to change quite considerably, it may be appropriate to point out that we study the relationships between certain subalgebras of an algebra A on the one hand, and certain groups of automorphisms of A on the other. The ordinary theory of central simple algebras covers the case of subalgebras containing the center of A. Another special case, in which the subalgebras are subfields of the center, has been discussed from a somewhat different point of view by Teichmueller [7] and by Eilenberg and MacLane [4](1). Some of the recent generalizations of Galois theory to far more general rings-in particular J. Dieudonne's [3]-are concerned chiefly with Galois correspondences between certain subrings of a given ring A on the one hand, and certain rings of endomorphisms on the other. These rings of endomorphisms are intimately related to groups of automorphisms of A in the case of division rings, but in the general case this theory is quite far removed from a theory of automorphisms of A. The theory given here resembles the Galois theory for division rings as given by H. Cartan [2] and N. Jacobson [6], and covers the special case of division algebras in the same way. Some of our results are related also to those of Dieudonne [3]. However, our method is independent of the new techniques and consists simply in combining the well known results of the theory of simple algebras with those of the ordinary Galois theory for fields, a knowledge of which we shall assume(2). 1. Auxiliary results.

On unicoherent continua

Abstract: The first section of this paper deals mainly with hereditarily unicoherent continua which are atriodic. The term \"continuum\" will imply compactness. A continuum M is said to be unicoherent if the set of all points common to any two continua whose sum is M is connected. It is said to be atriodic if it does not contain three continua such that the common part of each pair is the common part of all three and is a proper subcontinuum of each. A property of a continuum which is possessed by all its subcontinua is said to be hereditary. The second section is chiefly concerned with conditions implying hereditary unicoherence. Among other things it is proved that in order that a hereditarily decomposable continuum be hereditarily unicoherent it is necessary and sufficient that it contain no continuum N which can be decomposed into an upper semi-continuous collection of mutually exclusive continua with respect to the elements of which N is a simple closed curve. In establishing this theorem it is proved that a hereditarily decomposable irreducible continuum M can be decomposed into an upper semi-continuous collection with respect to the elements of which M is an arc. The third section is devoted to properties of a continuum every subcontinuum of which is unicoherent and decomposable which are analogous to certain properties of an acyclic continuous curve. The basic space is a Moore space and, since all the theorems deal with internal properties of continua, may therefore be described as metric. The reader is referred to R. L. Moore's Foundations of point set theory(l) for definitions of terms not defined here. 1. Atriodic hereditarily unicoherent continua.