Publication Date

7-17-1972

Abstract

Let G be a compact group and its dual group which we suppose to be countable. We suppose that S = {n}n≥0 is a non-decreasing sequence of finite subsets of having the property that = . If F is the topological dual of a homogeneous Banach space B provided with its norm topology, one can then define in a natural way convergence in norm in F with respect to the sequence S. With F one can associate two Banach spaces Fb and Fc consisting of those elements of F whose Fourier series has bounded partial sums and whose Fourier series converges in norm, respectively. In particular, when F = C(G), the space of continuous functions on G, Fc is usually denoted by U(G) or by U(G; S), and consists of those continuous functions on G whose Fourier series converge uniformly, the sequence of partial sums being taken with respect to the sequence S. It is then shown that for any F as above, the dual of Fc can be identified with the multipliers of Fc into U. In Chapter 3 the following problem is considered: let be a subset of and suppose that E is another Banach space of the same type as F such that F ⊂ E and the canonical injection F E is continuous, then what conditions must satisfy in order that ? Here Ẽ means the closure in E of the set of elements of E having only finitely many non-zero Fourier coefficients, and means the set of elements of E having Fourier coefficients equal to zero outside of . Theorem 1.2.1 gives an analytic characterization of such sets, which we denote by (E, Fb;, S). Immediately following the theorem many examples are given: among them (p) sets (E = M(T), F = Lp(T), S = symmetric partial sums), UN sets (E = F = L∞(T)) and R sets (E = F = M(T)). The rest of the thesis is concerned with UN sets and (p) sets. In Sections 3 and 4 of Chapter 3 an example of Figà-Talamanca is generalized and other examples of UN sets are given which shed some light on general questions of the theory. Finally, in Section 5, a structural property of UN sets is given, namely such sets cannot contain arbitrarily long arithmetic progressions.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Liang-Shin Hahn

Second Committee Member

Julius Rubin Blum

Third Committee Member

Arthur Steger

Language

English

Document Type

Dissertation

Download

Share

COinS