Bases properties of sequences of dilated periodic functions in Banach and Hilbert spaces

Abstract

The main aim of the thesis is to continue the investigations as to which extent the family of dilations Ef := {fn}n, where f : R→C and fn(·) := f(n·) for all n ∈N, forms a basis of Lr(0,1) for r ∈ (1,∞). We introduce an improved one-term and new multi-term criteria for determining Schauder and Riesz bases properties of the family Ef in the context of Lebesgue spaces. We develop the concept of multipliers on Hardy spaces of polydiscs and establish an analogy to the preceding criteria in this setting. We illustrate the rich structure behind this problem by applying these criteria to various families of generalised (p,q)-trigonometric functions, such as, the p-cosine, the p-sine, the p-exponential and the (p,q)-cosine functions. These functions arise naturally in the study of eigenspaces of the one-dimensional Dirichlet problem for the (p,q)-Laplacian. The approach was proved fruitful and the findings achieved follow naturally from previously known results.