Biorthogonal local trigonometric bases (Q2702480)

A local trigonometric basis consists of cosines and sines multiplied with a smooth well-localized window function. The purpose is to obtain a basis with good time-frequency localization, properties which are essential in applications like signal processing. The paper gives a review of known results, for example about the Coifman-Meyer bases and the biorthogonal bases by Jawerth/Sweldens and Chui/Shi. Another way to obtain a localized basis is to use a Gabor family \(\{e^{2\pi imbx}g(x-na)\}_{m,n\in Z}\) with a well-localized window \(g\in L^2(R)\). However, the Balian-Low theorem states that if \(\{e^{2\pi imbx}g(x-na)\}_{m,n\in Z}\) is a Riesz basis, then NEWLINE\[NEWLINE\int x^2|g(x)|^2dx \int \gamma^2 |\widehat{g}(\gamma)|^2d\gamma=\infty,NEWLINE\]NEWLINE i.e., \(g\) has poor time-frequency localization. Wilson bases were introduced in order to avoid this problem. The paper gives a new approach to biorthogonal Wilson bases, and relates them to the bases by Chui and Shi.NEWLINENEWLINEFor the entire collection see [Zbl 0954.65001].