Irregular wavelet frames and Gabor frames (Q2778248)

Let \(g \in L^2(R^d)\) and \(b>0\). The standard wavelet and Gabor systems are defined as the collections of functions: NEWLINE\[NEWLINE\{a^{jn/2} g (a^j x - bk): j \in Z,\;k \in Z^n \} \quad\text{and}\quad \{e^{2\pi i bj\cdot x} g(x - k): j,k \in Z^n \},NEWLINE\]NEWLINE respectively. It is of interest to study necessary and sufficient conditions for such systems to form orthonormal bases, frames, or complete systems [cf. \textit{E. Hernández, D. Labate} and \textit{G. Weiss}, ``A unified characterization of reproducing systems generated by a finite family. II'', J. Geom. Anal. 12, No. 4, 615-662 (2002)]. NEWLINENEWLINENEWLINEIn this paper the authors consider irregular wavelet and Gabor systems: NEWLINE\[NEWLINE\{ \lambda_j^{n/2} g (\lambda_j x - bk): j \in Z,\;k \in Z^n \}\quad \text{and}\quad \{e^{2\pi i bj\cdot x} g(x - \lambda_k): j,k \in Z^n \},NEWLINE\]NEWLINE where \(\{\lambda_j: j\in Z\} \subseteqq R^+\) and \(\{ \lambda_j: k\in Z\} \subseteqq R^{n}\). Different types of sufficient conditions are provided for such systems to form frames in \(L^2(R^n)\). Some of these conditions are related to Daubechies' autocorrelation conditions; others require certain growth assumptions on the collection \(\{\lambda_j: j \in Z\}\) or on the Fourier transform of the generating function \(g\). Examples are given of collections of dilations which do not yield wavelet frames.