Gabor Representations and Wavelets.

Abstract

Our modest goal in this paper is to define a generalization of the Fourier transform of L^1(R) and to prove the L^1(R) norm inversion theorem for such a transform (Theorem 1.5). Wiener's notion of deterministic autocorrelation (from the late 1920s) arises naturally in the proof of this result; and Gabor's representation of signals (from 1946), which is a fundamental example of wavelet decomposition, provides the setting.