Frames, Riesz bases, and discrete Gabor/wavelet expansions

This paper is a survey of research in discrete expansions over the last 10 years, mainly of functions in $L^2(\mathbb R)$. The concept of an orthonormal basis $\{f_n\}$, allowing every function $f \in L^2(\mathbb R)$ to be written $f=\sum c_nf_n$for suitable coefficients $\{c_n\}$, is well understood. In separable Hilbert spaces, a generalization known as frames exists, which still allows such a representation. However, the coefficients $\{c_n\}$ are not necessarily unique. We discuss the relationship between frames and Riesz bases, a subject where several new results have been proved over the last 10 years. Another central topic is the study of frames with additional structure, most important Gabor frames (consisting of modulated and translated versions of a single function) and wavelets (translated and dilated versions of one function). Along the way, we discuss some possible directions for future research.