Continuous Complex-valued Functions Research Articles

On uniformly approximable Sidon sets

Let G be a compact abelian group and let T be the character group of G. Suppose £ is a subset of T. A trigonometric polynomial f on G is said to be an ^-polynomial if its Fourier transform / vanishes off E. The set E is said to be a Sidon set if there is a positive number B such that 2^xeb |/(X)| a-B||/||u for all E-polynomials /; here, ||/||„ = sup{ |/(x)| : xEG}. In this note we shall discuss a certain class of Sidon sets—the class of all approximate Sidon sets. A Sidon set E is said to be uniformly approximable if the characteristic function of E is in the uniform closure of the algebra of FourierStieltjes transforms of Radon measures on G. The principal result in this note is a theorem which gives a list of characterizations of uniformly approximable Sidon sets. These are somewhat akin to characterizations for Sidon sets given by Hewitt and Zuckerman [2, 2.1 and 8.5] and by Rudin ([7] and [6, 5.7]). Also, we shall propose several alternative characterizations and show (except in one case) that these would-be characterizations are false. Steckin [8, p. 394] established a sufficient condition for Sidon sets, which has been generalized successively by Hewitt and Zuckerman [2], Rudin [7], and Rider [5]. Evidently all known Sidon sets are finite unions of Sidon sets which satisfy Rider's condition and it will be shown that all of these finite unions are uniformly approximable Sidon sets. Ramirez [3] has studied subsets of T whose characteristic functions are in the uniform closure of the algebra of Fourier-Stieltjes transforms. Among the results presented in [3] are several theorems about uniformly approximable Sidon sets. We must establish certain conventions about notation. Throughout the paper G denotes an arbitrary compact abelian group and T denotes the character group of G. We denote the measure algebras of G and T by M{G) and M{T) respectively and the group algebra of Haar-integrable functions on G by Z-i(G). We denote by C{G) the algebra of all complex-valued continuous functions on G. If A QT, then A' is the complement of A, %A is the characteristic function of A in T, and C0{A) denotes the set of all complex-valued functions on A which vanish at infinity. For A C.DQT, we let C{D, A) be the set of

Integral representation of multiplicative, involution preserving operators in ℒ(𝒞(𝒮),ℰ)

1. Introduction. Throughout, S will be a compact Hausdorff space and E will be a Banach space which is the dual of another Banach space F. C(S) will denote the space of complex-valued continuous functions on S topologized with the topology of uniform convergence. ?(C(S), E) will denote the space of continuous linear operators from C(S) to E. A theorem of Gil de Lamadrid [5, p. 103] identifies ?(C(S), E) with a space of E-valued measures, the correspondence between operator and measure being given by integration. A closely related result was given earlier by Bartle, Dunford, and Schwartz [2 ]. Now if E is a Banach algebra with involution [7, p. 178], it makes sense to consider operators which are not only continuous and linear but which also preserve multiplication and involution. A natural question arises: How are these additional properties reflected in the representing measure? We answer this question under additional restrictions on E. We also give several examples of spaces E satisfying the hypotheses of our theorems. One can use the results of this paper to prove the Spectral Theorem for bounded operators; but the proof follows standard lines and will not be included (see [6, p. 99]). We conclude this introduction with a precise description of Gil de Lamadrid's Theorem. Our description differs from Gil de Lamadrid's, but it is not difficult to verify that they are equivalent. We consider the class N(S, E) of set functions m from 6((S), the Borel class of S, to E which are countably additive and regular with respect to the weak topology a(E, F) on E induced by F. To say that m: (3(S) -E is regular with respect to the topology o(E, F) means that, for every BE@(S) and every a(E, F)-neighborhood N of 0, there exists a compact set K and an open set U such that KCBC U and, if A C U-K, then m(A) EN. Defining addition and scalar-multiplication in N(S, E) in the usual set-wise fashion; i.e., (m1+m2)(B)=m1(B) +m2(B) and (am,)(B) ==am(B), N(S, E) is a vector space. The following formula defines a norm on N(S, E) making it a Banach space:

A Note on Strongly Regular Function Algebras

Let A be a uniformly closed subalgebra of C(X), the algebra of all complex-valued continuous functions on a compact Hausdorff space X. If A separates the points of X and contains the constant functions, A is called a function algebra. The algebra A is said to be strongly regular on X if it has the following property.Property. For each f in A, each point x in X, and every , there is a neighbourhood U of x and a function g in A with g(y) = f(x) for all y in U and for all y in X.That is, each function in A is uniformly approximate on X by functions in A which are constant near any point of X. Stated in terms of ideals, strong regularity means that, for each x, the ideal of functions vanishing in a neighbourhood of x is uniformly dense in the maximal ideal at x.

A Characterization of the Algebra of Functions Vanishing at Infinity

1. In this paper, X will always denote a locally compact Hausdorff space, C0(X) the algebra of all complex-valued continuous functions vanishing at infinity on X and B(X) the algebra of all bounded continuous complex-valued functions defined on X. If X is compact, C0(X) is identical to B (X) and all the results of this paper are obvious. Therefore, we will assume at the outset that X is not compact. If A represents an algebra of functions, AR will denote the algebra of all real-valued functions in A.

Hilbert-Schmidt representations of groups

Introduction. A completely continuous representation of a locally compact group G is a unitary representation T of G for which the operator Tb is completely continuous whenever b is an element of the group C*-algebra C*(G). We have also that, with respect to the norm operator topology, T[L(G)1 is dense in the set T[L1(G)] which in turn is dense in the set T[cC(G)]. (Here is the linear space of all continuous complex-valued functions on G with compact support.) Further, it is known that, if T is an irreducible completely continuous representation of G, the set T[C*(G)] is precisely the algebra of all completely continuous operators on the Hilbert space H(T). (See [4].) Investigation of completely continuous representations has been fruitful because, first of all, such representations are more easily analyzed, and secondly, a large class of groups (CCR-groups [2]) satisfies the condition that each of their irreducible unitary representations is a completely continuous representation. The algebra A of all Hilbert-Schmidt operators on a Hilbert space is dense in the algebra of all completely continuous operators on that space. Therefore, it was natural to inquire what simplifications arise if one requires that a completely continuous representation of a group G should map either the set L(G) into the space of all HilbertSchmidt operators or the set L1(G) into the space of all HilbertSchmidt operators. In this paper we show that too much simplification occurs in the latter case. In fact, if Tf is a Hilbert-Schmidt operator wheneverf is an element of L'(G), then T must be finite dimensional. (See the theorem below.) In the first case mentioned above, the simplifications, if there are any, are more elusive, and we shall return to this question at the end of the article. DEFINITION 1. Let G be a locally compact group. By a unitary representation of G we mean a homomorphism T of G into the group of unitary operators on a Hilbert space H(T) which is continuous with respect to the weak-operator topology. (See [31.) We single out this definition from representation theory because, in the course of the proof below, the continuity condition here is heavily exploited.

The set of curvilinear convergence of a continuous function defined in the interior of a cube

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.

Boundary values of analytic functions. II

It is known that a complex-valued continuous functionS(x) and a Schwartz distribution can both be extended to an analytic functionŜ(z) in the complex plane minus the support ofS. Conditions are given for the existence of limits\(\mathop {\lim }\limits_{\varepsilon \to 0 + } \hat S(x + i\varepsilon )\)Ŝ(x+ie), in the ordinary sense, at certain points of the support ofS, for the case in whichŜ(z) is the Cauchy representation. In this way we obtain “local” Plemelj and dispersion relations. Possible generalizations and applications are discussed.

A generalization of the Mikusinski operational calculus

0. Introduction. In his version of the operational calculus Mikusinski uses as a starting point the familiar theorem from algebra tlhat every integral domain can be embedded isomorphically in a field. He shows that the class of complex-valued continuous functions defined on [0, cO) forms an integral domain when addition and multiplication are taken to be pointwise addition and convolution of functions, respectively. The resulting field is called the field of Mikusinski operators. In the present paper we generalize Mikusinski's basic results by developing an operational calculus for strongly continuous functions defined on [0, cc) and having values in any Banach algebra with unit element. The Mikusinski operational calculus will then be a special case of our theory, obtained by taking the Banach algebra to be the complex numbers. The paper is divided into four parts. The first gives an algebraic construction; the second shows how the algebraic construction may be used to develop the operational calculus mentioned above; the third indicates some basic results which carry over from the (Mikusinski) operational calculus to this more general setting. Finally, in the last section, we generalize within the framework of the operational calculus a recent theorem of T. K. Boehme. We assume the reader is familiar with the basic notations and results in the books by Mikusinski [3] and Erdelyi [2].

Hyperbolic Mean Automorphic Functions

Let IR n be Euclidean n-space with generic element x = (x 1 . . . . , x.) and let C(]R") be the space of continuous complex-valued functions f : IR ~ ~ r under the topology of uniform convergence on compact subsets of IR ~. A function f in C(IR ~) is said to be mean periodic if the span of the ordinary translates of f is not dense in C(IR"). There is an extensive literature on the subject of mean periodic functions. Clearly, there is a corresponding notion of "mean automorphic" functions associated with each semigroup of mappings of IR ~ into IRL Let H be the group of homeomorphisms qJ that map IR n onto ]R" in the following special way:

Some Remarks on Analytic Representations and Products of Distributions

H. J. BREMERMANNt 1. Representation of distributions by analytic functions of one complex variable. This section is a brief summary; most of the proofs may be found in Bremermann [5]. Compare also Beltrami-Wohlers [1]. Terminology and notations are consistent with L. Schwartz [12]. Let T be a distribution in (&'(E1)), where E1 denotes the reals, or T C (O'L2); then the function T'(z) = (1/2ri)(T, 1/t -z) is an analytic function for Im z # 0 (Im z imaginary part of z). T(z) represents T in the following sense: T(x + iE) T(x iE) converges to T for E -> 0+ in the topology of (O'). We call T(z) the Cauchy representation of T. For T C (D'(El)) the function (t z)f1 is not a test function, T(z), in general, does not exist, but the following result holds: there exists a pair of functions f+(z) and f_(z), analytic in the upper and lower half-plane, respectively, such thatf+(x + iE) f(x ie) converges to T for e -> 0+ in the topology of (OD'). We call f+ the forward function, f_ the backward function and the pair an analytic representation of T. Analytic representations of the same distribution differ by at most an eintire function [14], [24]. If a pair of analytic functions represents T, then their complex derivatives represent T'. If the complement of the support of a distribution is nonempty, then for any analytic representation of T the forward and backward functions f+ and f_ are analytic continuations of each other and remain analytic on the complement of the support of T. Not every pair of functions, analytic in the upper and lower half-planes, respectively, represents a distribution: f+(z) = f_(z) = exp (1/z2) is a counter-example. A tempered function is a continuous complex-valued function oni El that grows at infinity no faster than a polynomial. For a tempered function f

The essential set of function algebras

Let X be a compact Hausdorff space and C(X) the Banach algebra of all complex-valued continuous functions on X under the sup-norm. By a function algebra A we mean a closed subalgebra of C(X) which separates points and contains the constant functions. Let S(A) denote the space of maxinmal ideals of A and let E(A) denote the essential set of A as defined by Bear [1], i.e. E(A) is the hull of the largest closed ideal of C(X) which is contained in A. We will obtain another characterization of the essential set in the case when S(A) =X and thereby obtain some results of a global nature from local hypotheses.

Analytic embedding and mean square approximation

1. Introduction. Our results concern the generalized Hp spaces derived from certain function algebras, particularly those spaces associated with analytic discs embedded in the carrier space. Such general Hp spaces are obtained by considering a uniform algebra =A(X), fixing a measure monX representing a multiplicative linear functional on A, and defining H(dm) as the closure of in Lv(dm) (w* closure for p= ). One further condition (the unique representing measure property) is imposed on so that a large portion of the classical theory concerning the Hardy classes remains valid for H*(dm). Our first result provides an explicit determination of the analytic embedding mapping of Wermer [7] (of the unit disc onto the Gleason part associated with ra) in terms of the outer function associated with a certain Radon-Nikodym derivative. We then use this explicit expression to generalize a result of Szegb' on mean square approximation of the function 1 by maximal ideals of functions, relating two different homomorphisms in the same Gleason part. 2. Preliminaries. If A is a compact Hausdorff space, a uniformly closed algebra A=A(X) of continuous complex-valued functions is

Representation of functions in 𝐶(𝑋) by means of extreme points

Let X be a compact metric space. It is known that if U is the closed unit ball of C,(X) (the space of continuous real-valued functions on X under the usual sup norm), a necessary and sufficient condition that U be the closed convex hull of the set of its extreme points is that X be totally disconnected (Bade [1]). It is also known (Phelps [4]) that if C(X) is the space of all continuous complex-valued functions on X under the sup norm, and if U is the closed unit ball of C(X), U is always equal to the closed convex hull of the set of its extreme points (see also Goodner [2]). It is our purpose in this note to obtain information about U in the case of C(X) similar to that obtained for Cr(X). We make the following notational conventions: D will denote the closed unit disc in the complex plane and B will denote the set of points in D of modulus 1. By E we will mean the set of extreme points of U (the closed unit ball of C(X)); E is the set of all elements of U which map X into B. The topological dimension of X as defined in Hurewicz and Wallman [3] will be denoted by dim X. Our theorem now reads as follows:

A theorem of Stone-Weierstrass type

The Stone-Weierstrass theorem gives very simple necessary and sufficient conditions for a subset A of the algebra of all real-valued continuous functions on the compact Hausdorff space X to generate a subalgebra dense in namely, this is so if and only if the functions of A strongly separate the points of X, in other words given any two distinct points of X there exists a function in A taking different values at these points, and given any point of X there exists a function in A non-zero there. In the case of the algebra of all complex-valued continuous functions on X, the same result holds provided that we consider the subalgebra generated by A together with Ā, the set of complex conjugates of the functions in A.

On the Boundary and Tensor Product of Function Algebras

Let be an arbitrary family of continuous complex-valued functions defined on a compact Hausdorff space X. A closed subset B ⊆ X is called a boundary for if every attains its maximum modulus at some point of B. A boundary, B, is said to be minimal if there exists no boundary for properly contained in B. It can be shown that minimal boundaries exist regardless of the algebraic structure which may possess. Under certain conditions on the family , it can be shown that a unique minimal boundary for exists. In particular, this is the case if is a subalgebra or subspace of C(X) where X is compact and Hausdorff (see for example [2]). This unique minimal boundary for an algebra of functions is called the Silov boundary of .

The maximal ideal space of functions locally approximable in a function algebra

Introduction. Let ft be a compact Hausdorff and C(Q) the Banach algebra of all complex-valued continuous functions on ft. Also let 21 be a closed subalgebra of C(ft) which separates the points of ft and contains the constants. We shall also assume that ft is the carrier or space of maximal of 21. This amounts to requiring that every homomorphism c/> of 21 onto the complex numbers be given by a point of ft [5, §3.1]. In other words, there is a point co,j such that d(<p) =a(co0) for every a£2l, where o(c6) denotes the image of a under the homomorphism c6. Obviously every point of ft determines such a homomorphism. A complex function/ defined in ft is said to be locally approximable in 21 if every point of ft has a neighborhood in which / is a uniform limit of elements of 21. We also call such functions %-holomorphic [4]. If every point of ft has a neighborhood in which / actually coincides with an element of 21, then / is said to belong locally to SI. The 21holomorphic functions are obviously continuous and contain the functions that belong locally to 21. The closure in C(ft) of the 21holomorphic functions will be denoted by $81 and the closure of the functions that belong locally to 21 will be denoted by $80Evidently SlCSoCgSjCCXft), and both 9}0 and 5Bi are subalgebras of C(Q). As is shown by an example of Eva Kallin's [2], it may happen that 33u5^2l. On the other hand, G. Stolzenberg [7] has proved that $80 nevertheless must have ft as its of maximal ideals. The object of this paper is to prove that the same conclusion holds for the algebra 93i. In fact we shall prove the more inclusive result that ft is the of maximal ideals for any closed subalgebra 93 of C(ft) which contains 2t and is generated by 2I-holomorphic functions. In particular, the Stolzenberg result is included. Stolzenberg's proof depends on the nontrivial fact that 21 satisfies a local maximum modulus principle [6, 6.1, p. 9]. Our proof, though quite different in spirit, also rests ultimately on the local maximum principle since it involves a result concerning convexity of abstract analytic varieties which we have obtained using this principle [4]. The theorem ob-

A generalized Banach-Stone theorem

xeX (We recall that if X is compact, then C0(X) consists of all continuous complex-valued functions on X.) The well known Banach-Stone theorem states that if X and Y are locally compact Hausdorff spaces, and if C0(X) and C0(Y) are isometrically isomorphic, then X and Y are homeomorphic. The purpose of this article is to show that, with the additional hypothesis of first countability (which may well be unessential), the conclusion of the theorem remains true for a class of mappings somewhat more general than isometries. We wish to prove the following: THEOREM. If X and Y are first countable, locally compact Hausdorff spaces, and 4 is a continuous, norm-increasing linear isomorphism of Co(X) onto Co(Y) with bound strictly less than two, lIf!! _ lf+(f)ff < 1lfif lIf!, f E CO(X), 1ffij < 2,

On uniform approximation by rational functions

If X is a compact subset of the complex plane, C(X) will denote the sup norm algebra of continuous complex-valued functions on X; R(X) will denote the uniformly closed subalgebra of C(X) generated by the rational functions with poles off X. Whenever X has an interior, R(X) is a proper subalgebra of C(X). Mergelyan [I ] has shown that it is still possible for R(X) to be strictly contained in C(X) even if X has no interior: R(X) C(X) whenever X is a set formed by deleting from the interior of the closed unit disc a sequence of open discs with mutually nonintersecting boundaries of finite total length in such a way as to leave no interior. In order to understand how this R(X) could be a proper subalgebra of C(X), it had been conjectured that any such R(X) must be antisymmetric, i.e., must contain no nonconstant real-valued function. This conjecture will be proved false by the construction of a counterexample. The strategy of construction is to mark off on the horizontal diameter of the unit disc a general Cantor set. The vertical lines determined by the endpoints of the intervals complementary to the Cantor set then serve as guidelines which will be covered almost everywhere by open discs to be deleted from the unit disc. In order to insure that the total length of the circumferences of the deleted discs is finite, it is necessary to alternate between selecting an interval complementary to the Cantor set, and selecting the discs which cover the associated pair of guidelines: at each stage we must select the interval in such a way (i.e., large enough) as to insure that a large portion of the length of the associated guidelines is already deleted by the discs which were selected at the preceding stages. The final step-that of showing that the algebra R(X) is not antisymmetric-involves proving that the Cantor function (extended as a constant in the vertical direction) is in R(X). This proof relies on a lemma due to Mergelyan [1], which may be stated in this form:

Translation-invariant function algebras on compact groups

Let X be a compact group. $(X) denotes the Banach algebra (point multiplication, sup norm) of continuous complexvalued functions on X A is any closed subalgebra of d(X) which is stable under right and left translations and contains the constants. It is shown, by means of the Peter-Weyl Theorem and some multilinear algebra, that the condition (*) every representation of degree 1 of X has finite image is necessary and sufficient that every possible A be self-ad joint. If X is connected, then (*) means that X is a projective limit of semisimple Lie groups; if X is a Lie group, then (*) means that X is semisimple. The Stone-Weierstrass Theorem then gives a quick classification of all possible algebras A on an arbitrary connected semisimple Lie group X.

On families of equi-uniformly distributed sequences in compact spaces

Let X be a compact Hausdorff space, let p be a regular normed Borel measure on X, and let C(X) be the Banach space (under uniform norm II1[I -sup I/(x)]) of all complex-valued continuous functions f on X. Identifying xEX measure and corresponding integral on C(X) we shall use the notation p(/) = ff(x)d/~(x). A family ~----((x~,n)~=l : a E S) of sequences in X is called x a/amily o 1 e~lui-/~-uni/ormly distributed sequences (in German: , , g le ichm~ /~gleichverteih) if for every f E C(X) and for every real number e > 0 there exists an integer N(/, e), indepedent of a, such that