zbMATH — the first resource for mathematics

A characterization of functions that generate wavelet and related expansion. (English) Zbl 0896.42022
Let \(\Phi=\{\phi^1,\ldots,\phi^L\}\) be a family of scaling functions in \(L^2({\mathbb{R}}^n)\) and define \(\phi_{k,{\mathbf k}}^\ell({\mathbf x})=2^{jn/2}\phi^\ell(2^j{\mathbf x}-{\mathbf k})\). Let \(\Psi=\{\psi^1,\ldots,\psi^L\}\) be a corresponding set of wavelets with \(\psi_{j,{\mathbf k}}^\ell\) defined similarly to \(\phi_{k,{\mathbf k}}^\ell\). This paper characterizes those families \(\Phi\) and \(\Psi\) such that for each \(f\in L^2({\mathbb{R}}^n)\) one has \(f=\sum_{\ell=1}^L\sum_{j\in{\mathbb{Z}}}\sum_{{\mathbf k}\in{\mathbb{Z}}^n} (f,\phi_{k,{\mathbf k}}^\ell)\psi_{k,{\mathbf k}}^\ell\) where \((\cdot,\cdot)\) is the inner product in \(L^2({\mathbb{R}}^n)\) and the convergence is with respect to the derived norm. The characterization is expressed as identities for certain series in the Fourier transform functions \(\widehat{\phi}^\ell\) and \(\widehat{\psi}^\ell\) which should hold a.e.

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
Full Text: DOI EuDML
[1] Auscher, P. (1995). Solutions to two problems on wavelets.J. Geometric Anal., 5(2), 181–237. · Zbl 0843.42015
[2] Chui, C.K. and Shi, X. (1993). Bessel Sequences and Affine Frames.Appl. Computational Harmonic Anal., 1(1), 22–49. · Zbl 0788.42011
[3] Dai, X., Larson, D.R., and Speegle, D.M. (1995). Wavelet sets in \(\mathbb{R}\) n . Preprint.
[4] Frazier, M. and Jawerth, B. (1985). The{\(\Phi\)}-transform and applications to distribution spaces.Springer Lecture Notes in Mathematics 1302, 223–246.
[5] Frazier, M., Jawerth, B., and Weiss, G.L. (1991). Littlewood – Paley Theory and the study of function spaces.AMS Regional Conf. Ser. Math., 79. · Zbl 0757.42006
[6] Gripenberg, G. (1995). A necessary and sufficient condition for the existence of a father wavelet.Studia Math., 114(3), 207–226. · Zbl 0838.42012
[7] Kahane, J.P. and Lemarié-Rieusset, P.G. (1995).Fourier Series and Wavelets. Gordon & Breach Publishers. · Zbl 0966.42001
[8] Hernández, E. and Weiss, G.L. (1996).A First Course on Wavelets. CRC Press, Boca Raton, FL. · Zbl 0885.42018
[9] Meyer, Y. (1992).Wavelets and Operators. Cambridge University Press. · Zbl 0776.42019
[10] Songer, I. (1970).Bases in Banach Spaces I. Springer-Verlag.
[11] Soardi, P. and Weiland, D. (1996). MSF wavelets in several dimensions. Preprint.
[12] Wang, X. (1995) The Study of Wavelets from the Properties of their Fourier Transforms. Ph.D. Thesis, Washington University.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.