A construction of multiresolution analysis by integral equations

Lee, Dong-Myung; Lee, Jung-Gon; Yoon, Sun-Ho

doi:10.1090/S0002-9939-02-06713-8

A construction of multiresolution analysis by integral equations
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by Dong-Myung Lee, Jung-Gon Lee and Sun-Ho Yoon PDF

Proc. Amer. Math. Soc. 130 (2002), 3555-3563 Request permission

Abstract:

In this paper we present a versatile construction of multiresolution analysis of two variables by means of eigenvalue problems of the integral equation, for $\lambda =2$. As a consequence we show that if $\phi (x)$ is the solution of the equation $\phi (x) = \lambda \int _{\mathbb {R}} h (2x-y) \phi (y)dy$ with $supp \hat h(\omega ) = [-\pi , \pi ]$, then $V_{j} =span \{ \phi (2^{j} x_{1} -k_{1} )$ $\phi (2^{j} x_{2} -k_{2} ) | k_{1} , k_{2} \in \mathbb {Z} \}$ constructs a two-variable multiresolution analysis.

References

J. Aguirre, M. Escobedo, J. C. Peral, and Ph. Tchamitchian, Basis of wavelets and atomic decompositions of $H^1(\textbf {R}^n)$ and $H^1(\textbf {R}^n\times \textbf {R}^n)$, Proc. Amer. Math. Soc. 111 (1991), no. 3, 683–693. MR 1050015, DOI 10.1090/S0002-9939-1991-1050015-2
Peter G. Casazza and Ole Christensen, Frames containing a Riesz basis and preservation of this property under perturbations, SIAM J. Math. Anal. 29 (1998), no. 1, 266–278. MR 1617185, DOI 10.1137/S0036141095294250
Charles K. Chui, An introduction to wavelets, Wavelet Analysis and its Applications, vol. 1, Academic Press, Inc., Boston, MA, 1992. MR 1150048
M. G.Cui, D. M. Lee, and J. G. Lee, Fourier Transforms and Wavelet Analysis, Kyung Moon Press (2001).
Ronald R. Coifman and Yves Meyer, Remarques sur l’analyse de Fourier à fenêtre, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 3, 259–261 (French, with English summary). MR 1089710
Ingrid Daubechies and Jeffrey C. Lagarias, Two-scale difference equations. I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991), no. 5, 1388–1410. MR 1112515, DOI 10.1137/0522089
Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107, DOI 10.1137/1.9781611970104
K. Gröchenig and W. R. Madych, Multiresolution analysis, Haar bases, and self-similar tilings of $\textbf {R}^n$, IEEE Trans. Inform. Theory 38 (1992), no. 2, 556–568. MR 1162214, DOI 10.1109/18.119723
A. Grossmann, J. Morlet, and T. Paul, Transforms associated to square integrable group representations. I. General results, J. Math. Phys. 26 (1985), no. 10, 2473–2479. MR 803788, DOI 10.1063/1.526761
Björn Jawerth and Wim Sweldens, An overview of wavelet based multiresolution analyses, SIAM Rev. 36 (1994), no. 3, 377–412. MR 1292642, DOI 10.1137/1036095
Wayne M. Lawton, Necessary and sufficient conditions for constructing orthonormal wavelet bases, J. Math. Phys. 32 (1991), no. 1, 57–61. MR 1083085, DOI 10.1063/1.529093
J. G. Lee and D. M. Lee, A Filtering Formula on Wavelets, Korean Annales of Math. 15 (1998), 247–255.
S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation, IEEE Trans. Pattern Anal. Machine Intell. 11 (1989).