Constructing pairs of dual bandlimited framelets with desired time localization
The author constructs dual families of framelets in which one of the families is generated by a bandlimited wavelet \(\psi\) whose dilates form a partition of unity of \(\mathbb{R}\) while the dual generator \(\widetilde\psi\) is a linear combination of dilates of \(\psi\) whose coefficients can be designed such that \(\widetilde{\psi}\) has desired properties. The main result is stated precisely as follows. Let \(n\) be a positive integer, let \(a>1\), and let \(\psi\in L^2(\mathbb{R})\) be such that the Fourier transform \(\widehat{\psi}\) is a bounded, real-valued function supported in \([-a^p,-a^{p-n}]\cup [a^{p-n},a^p]\) for some integer \(p\) and such that \(\sum_{j=-\infty}^\infty \widehat{\psi}(a^j\xi)=1\) almost everywhere. Let \(b\in [0, 1/(2a^p)]\). Then \(\{D^j_a T_{bk} \psi\}\) generates a wavelet frame for \(L^2(\mathbb{R})\). Here, \(D_a \psi(x)=\sqrt{a} \psi(ax)\) and \(T_b\psi(x)=\psi(x-b)\). In addition, if \[ \phi(x) = b\psi(x)+2b\sum_{j=1}^{n-1} a^{-j} \psi(x/a^j) \] then \(\{D^j_a T_{bk} \phi\}\) generates a dual frame for \(L^2(\mathbb{R})\). After providing several specific examples of \(\psi\) satisfying the hypotheses, it is shown that the ranges of the translation parameter and of the scales used in defining \(\phi\) can be extended, and some alternative recipes for a dual generator are proposed. More fundamentally, the author provides a rationale for circumstances in which the dual generator \(\phi\) might be more useful than the canonical frame dual.
scientific article; zbMATH DE number 5549577