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 (R). The concept of an orthonormal basis {f(n)}, allowing every function f is an element of L-2 (R) to be written f = Sigma c(n)f(n) for suitable coefficients {c(n)}, is well understood. In s eparable Hilbert spaces, a generalization known as frames exists, which sti ll allows such a representation. However, the coefficients {c(n)} are not n ecessarily unique. We discuss the relationship between frames and Riesz bas es, a subject where several new results have been proved over the last 10 y ears. Another central topic is the study of frames with additional structur e, most important Gabor frames (consisting of modulated and translated vers ions of a single function) and wavelets (translated and dilated versions of one function). Along the way, we discuss some possible directions for futu re research.