Pseudoframe multiresolution structure on abelian locally compact groups

Document Type: Research Paper


1 Ph. D. student in Ferdowsi University of Mashhad

2 Department of pure Mathematics; Ferdowsi University of Mashhad;

3 Department of pure Mathematics;Ferdowsi University of Mashhad;


‎Let $G$ be a locally compact abelian group‎. ‎The concept of a generalized multiresolution structure (GMS) in $L^2(G)$ is discussed which is a generalization of GMS in $L^2(mathbb{R})$‎. ‎Basically a GMS in $L^2(G)$ consists of an increasing sequence of closed subspaces of $L^2(G)$ and a pseudoframe of translation type at each level‎. ‎Also‎, ‎the construction of affine frames for $L^2(G)$ based on a GMS is presented‎.


[1] J. J. Benedetto, S. Li, The theory of multiresolution analyses frames and applications to filter banks, Appl. Comp. Harm. Anal., 5 (1998), 389-427.
[2] J. J. Benedetto, S. Li, Multiresolution analysis frames with applications, Proceeding ICASSP93 Proceedings of IEEE international conference on Acoustics, speech, and signal processing: digital speech processing Volume III Pages 304-307, 1993.
[3] P. G. Casazza O. Christensen, D. Stoeva, Frame expansions in separable Banach space, J. Math. Anal. Appl., 114(1) (2005), 710–723.
[4] O. Christensen, On frame multiresolution analysis, In An Introduction to Frames and Riesz Bases, Part of the series Applied and Numerical Harmonic Analysis, 283-311, 2003.
[5] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, Boston 2003.
[6] S. Dahlke, Multiresolution Analysis and Wavelets on Locally Compact Abelian Groups, Wavelets, Images, and Surface Fitting. P. J. Laurent, A. Le Mehaute, L. L. Schumaker, eds., A. K. Peters, Wellesley, 1994, 141-156.
[7] I. Daubechies, Orthonormal bases of compactly supported wavelets, Commun. Pure Appl. Math., 41(9) (1988), 909-996.
[8] I. Daubechies, A. Grossmann, Y. Meyer, Painless nonorthogonal expansion, J. Math. Phys. 27 (1986), 1271– 1283.
[9] R. Duffin R, S. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), 341–366.
[10] Yu. A. Farkov, Orthogonal wavelets on locally compact abelian groups, Funktsional. Anal. i Prilozhen., 31(4) (1997), 86-88, English transl., Funct. Anal. Appl., 31 (1997), 294-296.
[11] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 1995.
[12] D. Gabor, Theory of communications, J. Inst. Electr. Eng., 93(26) (1946), 429–457.
[13] R. A. Kamyabi Gol, R. Raisi Tousi, The structure of shift invariant spaces on a locally compact abelian group, J. Math. Anal. Appl., 340(1) (2008), 219–225.
[14] R. A. Kamyabi Gol, R. Raisi Tousi, Some equivalent multiresolution conditions on locally compact abelian
groups, Proc. Math. Sci., 120(3) (2010), 317–331.
[15] S. V. Kozyrev, Wavelet theory as p-adic spectral analysis, Izv. Ross. Akad. Nauk Ser. Mat., 66(2) (2002), 149-158, English transl, Izv. Math., 66 (2002), 367–376.
[16] W. C. Lang, Wavelet analysis on the Cantor dyadic group, Houston J. Math., 24(3)(1998), 533–544.
[17] S. Li, The theory of frame multiresolution analysis and its applications, Ph. D. Thesis, University of Maryland Graduate School, Baltimore, May 1993.
[18] S. Li, A theory of generalized multiresolution structure and pseudoframes of translates, J. Fourier Anal. and Appl. 7(1) (2001), 23–40.
[19] S. Li, H. Ogawa, A theory of peseudoframes for subspaces with applications, Proc. SPIE 3458, Wavelet Applications in Signal and Imaging Processing VI, 67(1998); doi:10.1117/12.328126.
[20] S. Li, H. Ogawa, Pseudoframes for subspaces with applications, J. Fourier Anal. Appl., 10(4)(2004), 409–431.
[21] S. G. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2(R), Trans. Amer. Math. Soc., 315(1) (1989), 69–87.
[22] Y. Meyer, Wavelets and Operators, Translated by DH Salinger, Cambridge Studies in Advanced Mathematics, 1992.
[23] D. P. Petersen, D. Middleton, Sampling reconstruction of wave-number limited functions in N-dimensional Euclidean spaces, Inf. Control, 5(4)(1962), 279–323.