Dual wavelets associated with nonuniform MRA

Main Article Content

Mohd Younus Bhat


A generalization of Mallats classical multiresolution analysis, based on thetheory of spectral pairs, was considered in two articles by Gabardo and Nashed. In thissetting, the associated translation set is no longer a discrete subgroup of R but a spectrumassociated with a certain one-dimensional spectral pair and the associated dilation is aneven positive integer related to the given spectral pair. In this paper, we construct dualwavelets which are associated with Nonuniform Multiresolution Analysis. We show thatif the translates of the scaling functions of two multiresolution analyses are biorthogonal,then the associated wavelet families are also biorthogonal. Under mild assumptions onthe scaling functions and the wavelets, we also show that the wavelets generate Rieszbases

Article Details

How to Cite
Bhat, M. Y. (2021). Dual wavelets associated with nonuniform MRA. Tamkang Journal of Mathematics, 50(2), 119–132. https://doi.org/10.5556/j.tkjm.50.2019.2646
Author Biography

Mohd Younus Bhat, Department of Mathematics National Institute of Technology Srinagar, Jammu and Kashmir, India.

Department of Mathematics National Institute of Technology Srinagar, Jammu and Kashmir, India.


B. Behera and Q. Jahan, Biorthogonal wavelets on local fields of positive characteristic, Comm. Math. Anal.,

(2013), 52–75.

M. Bownik and G. Garrigos, Biorthogonal wavelets, MRA’s and shift-invariant spaces, Studia Mathematica,

(2004), 231–248.

A. Calogero and G. Garrigos, A characterization of wavelet families arising frombiorthogonalMRA’s ofmulti-

plicity d, J. Geom. Anal., 11 (2001), 187–217.

A. Cohen and I. Daubechies, A stability criterion for biorthogonal wavelet bases and their related subband

coding scheme, Duke Math. J., 68 (1992), 313-335.

A. Cohen, I. Daubechies and J. C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl.Math., 45 (1992), 485–560.

L. Debnath and F. A. Shah,Wavelet Transforms and Their Applications, Birkhäuser, New York, 2015.

I. Daubechies, Ten Lectures onWavelets, SIAM: Philadelphia, 1992.

M. Frazier, G. Garrigos, K. Wang and G. Weiss, A Characterization of functions that generate wavelet and

related expansions, J. of Fourier Ana. and Appl., 3 (1997), 883–906.

J. P. Gabardo andM. Z. Nashed, Nonuniform multiresolution analysis and spectral pairs, J. Funct. Anal., 158(1998), 209–241.

J. P. Gabardo and X. Yu, Wavelets associated with nonuniform multiresolution analysis and one-dimensional

spectral pairs, J.Math. Anal. Appl., 323(2006), 798–817.

E. Hernanez, X.Wang, G. Weiss, Smoothing minimally supportes frequency wavelets. I, J. Fourier Anal. Appl.,

(1996), 329–340.

E. Hernanez, X.Wang, G.Weiss, Smoothingminimally supportes frequency wavelets. II, J. Fourier Anal. Appl.,

(1996), 23–41.

H. O. Kim, R. Y. Kim and J. K. Lim, Characterizations of biorthogonal wavelets which are associated with

biorthogonalmultiresolution analyses, Appl. Comput. Harmon. Anal., 11 (2001), 263–272.

R. Long and D. Chen, Biorthogonal wavelet bases on Rd , Appl. Comput. Harmon. Anal., 2 (1995), 230–242.

F. A. Shah and Abdullah, Nonuniform multiresolution analysis on local fields of positive characteristic, Comp.

Anal. Opert. Theory., 9(2015), 1589–1608.

F. A. Shah andM. Y. Bhat, Vector-valued nonuniform multiresolution analysis on local fields, Int. J. Wavelets,

Multiresolut. Inf. Process., 13(2015), No.4, Article Id: 1550029.132 MOHAMMAD YOUNUS BHAT

F. A. Shah and M. Y. Bhat, Nonuniform wavelet packets on local fields of positive characteristic, Filomat, 6

(2017), 1491–1505.