Linear Independence and Series Expansions in Function Spaces

Abstract

the representation of v in terms of the basis vectors is unique. In this article we consider complex vector spaces generated by certain special func tions and examine whether their linear combinations have unique representations. We start with the most elementary case, namely, trigonometric functions. After that we consider complex exponential functions and other more complicated systems of func tions (Gabor systems and wavelet systems) that have recently attracted much attention both in pure mathematics and in applied science. We present some open problems re lated to those systems, problems that are easy to formulate but apparently very difficult to solve. Finally, we introduce frames, which generalize the concept of an orthonormal basis. The motivation for this generalization comes from Gabor analysis, where we show that certain desirable properties are incompatible with the orthonormal basis re quirement. We show how the concept of linear dependence for wavelet systems plays a key role in modern constructions of frames having wavelet structure.