T. W. Gamelin

A theorem on polynomial-star approximation

We prove that the unit ball of a Banach space is polynomial-star dense in the unit ball of its bidual. This strengthens Goldstines theorem on weak-star density.

Pointwise bounded approximation and Dirichlet algebras

Abstract Those open sets U of S2 for which A(U) is pointwise boundedly dense in H∞(U) are characterized in terms of analytic capacity. It is also shown that the real parts of the functions in A(U) are uniformly dense in CR(∂U) if and only if each component of U is simply connected and A(U) is pointwise boundedly dense in H∞(U).

Analytic functions on Banach spaces

In these lectures we discuss various topics related to algebras of analytic functions on Banach spaces. We begin in Chapter 1 with a brief development of function theory on a Banach space, and we prove Ryan’s theorem that the Dunford-Pettis property implies the polynomial Dunford-Pettis property. Chapter 2 is devoted to extensions of analytic functions to the bidual. It includes a proof of the Aron-Herves-Valdivia theorem. Chapter 3 is devoted to approximation by finite-type polynomials, including theorems of Littlewood and Pitt. In Chapter 4 we study the algebra of entire functions that are bounded on bounded sets, and in Chapter 5 we take a brief look at the algebra of bounded analytic functions on the open unit ball of a Banach space.

Constructive techniques in rational approximation

This paper consists of several loosely organized remarks on the constructive methods for rational approximation developed by Vitushkin in [13]. These remarks are grouped under three headings. The first topic, taken up in ?1, illustrates the simplest case of the approximation scheme, and shows how it can be used to give simple proofs of rational approximation theorems on a class of infinitely connected compact sets. Special cases of these theorems have been proved by other methods by Fisher [8] and Zalcman [16]. This section also serves as an introduction to the integral operator TO, which is used in ??2 and 3. The second topic, involving pointwise bounded approximation, occupies ??2 through 5. In ?2, necessary and sufficient conditions are given on an open plane set U in order that every bounded analytic function on U be a pointwise limit on U of a bounded sequence of uniformly continuous analytic functions on U. This result, together with Mergelyans theorem, yields the Farrell-Rubel-Shields theorem (cf. [12]) on pointwise bounded approximation by polynomials, and its extension to finitely connected sets (cf. Corollary 3.3) by Ahern and Sarason [1]. It will be noted, however, that the constructive techniques alone do not give the best possible bounds on the norms of the approximating functions. In ?3 we obtain partial results on pointwise bounded approximation by rational functions. ??4 and 5 are devoted to constructing an example of a set for which pointwise bounded approximation by uniformly continuous analytic functions obtains, whereas approximation by rational functions fails. The final topic, relegated to ?6, involves extending Vitushkins techniques to vector-valued functions, in order to obtain results on uniform approximation by analytic functions of several complex variables. A related result has been given by

Tight uniform algebras and algebras of analytic functions

Abstract A uniform algebra A on a compact space X is tight if for each g ϵ C(X), the Hankel-type operator f → gf + A from A to C A is weakly compact. Two families of uniform algebras are shown to be tight: the algebras such as R(K) that arise in the theory of rational approximation on compact subsets of the complex plane, and algebras of analytic functions on domains in C n for which a certain ∂ -problem is solvable. A couple of characterizations of tight algebras are given, and one of these is used to show that the property of being tight places severe restrictions on the Gleason parts of A and the measures in A⊥.

Spectra of composition operators on algebras of analytic functions on Banach spaces

Let E be a Banach space, with unit ball B E . We study the spectrum and the essential spectrum of a composition operator on H ∞ ( B E ) determined by an analytic symbol with a fixed point in B E . We relate the spectrum of the composition operator to that of the derivative of the symbol at the fixed point. We extend a theorem of Zheng to the context of analytic symbols on the open unit ball of a Hilbert space.

Homomorphisms of Uniform Algebras

The aim of this paper is to describe some fairly recent results on homomorphisms of uniform algebras. It includes an exposition of results of Udo Klein obtained in his thesis, and some work of the author on algebras of analytic functions on domains in the plane.

Hankel Operators on Bounded Analytic Functions

For U a domain in the complex plane and g a bounded measurable function on U , the generalized Hankel operator Sg on H∞(U) is the operator of multiplication by g followed by projection into L∞/H∞. Under certain conditions on U we show that either Sg is compact or there is an embedded `∞ on which Sg is bicontinuous. We characterize those g’s for which Sg is compact in the case that U is a Behrens roadrunner domain.

A Stone-Weierstrass theorem for weak-star approximation by rational functions

Abstract Let K be a compact subset of the complex plane C, and let μ be a finite, positive Borel measure on K. Define R∞(μ) to be the weak-star closure in L∞(μ) of the algebra of rational functions with poles off K. For f ϵ R∞(μ), we consider A∞(f, μ), the weak-star closure in L∞(μ) of the algebra generated by R∞(μ) and the complex conjugate f of f. A problem which arises in connection with subnormal operators is to determine for which f ϵ R∞(μ), A∞(f, μ) = L∞(μ). We obtain a characterization of A∞(f,μ) as precisely the functions in L∞(μ) which belong to R∞(μE) for every restriction of μ to a level set E of f of positive measure. This characterization gives a solution to the problem above.

Arithmetic based fractals associated with Pascal's triangle

Our goal is to study Pascal-Sierpinski gaskets, which are certain fractal sets defined in terms of divisibility of entries in Pascal’s triangle. The principal tool is a “carry rule” for the addition of the base-q representation of coordinates of points in the unit square. In the case that q = p is prime, we connect the carry rule to the power of p appearing in the prime factorization of binomialcoefficients. We use the carry rule to define a family of fractal subsets Bqr of the unit square, and we show that when q = p is prime, Bqr coincides with the Pascal-Sierpinski gasket corresponding to N = pr . We go on to describe Bqr as the limit of an iterated function system of “partial similarities”, and we determine its Hausdorff dimension. We consider also the corresponding fractal sets in higher-dimensional Euclidean space.