Stability theorems for Fourier frames and wavelet Riesz bases

Abstract

In this paper we present two applications of a Stability Theorem of Hilbert frames to nonharmonic Fourier series and wavelet Riesz basis. The first result is an enhancement of the Paley-Wiener type constant for nonharmonic series given by Duffin and Schaefer in [6] and used recently in some applications (see [3]). In the case of an orthonormal basis our estimate reduces to Kadec' optimal 1/4 result. The second application proves that a phenomenon discovered by Daubechies and Tchamitchian [4] for the orthonormal Meyer wavelet basis (stability of the Riesz basis property under small changes of the translation parameter) actually holds for a large class of wavelet Riesz bases.

Frames, Riesz bases, and discrete Gabor/wavelet expansions

“…As the proof shows, 1 ||T † || 2 is a lower frame bound for {f n }. Actually, this is the optimal lower bound.…”

“…Combining the proof of Kadec's Theorem in [51] with the perturbation results for frames in [19], it is an easy matter to extend the result to frames; cf. [1], [15]. …”

“…We usually think about functions f ∈ L 2 (R) as signals. Recall that a basis {f n } for L 2 (R) allows every f ∈ L 2 (R) to be written f = n c n f n (1) for a unique set of scalar coefficients {c n }. Usually we think about (1) as a decomposition of the "complicated" signal f into a sum of the "simpler building blocks" f n that might be easier to work with.…”

An Introduction to Irregular Weyl-Heisenberg Frames

“…The idea is to transfer these calculations to L 2 [0, b] where we have many more results (especially perturbation theory [1,14]) to work with. This idea has not been fully explored yet and we offer it here as a tool for further study.…”

“…The importance of Theorem 5.4 is that it allows us to transfer Weyl-Heisenberg frame calculations from L 2 (R) to L 2 0, 1 b where we have very strong perturbation results of Balan [1] and Christensen [14].…”