Show Summary Details
More options …

# Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio

IMPACT FACTOR 2018: 0.551

CiteScore 2018: 0.52

SCImago Journal Rank (SJR) 2018: 0.320
Source Normalized Impact per Paper (SNIP) 2018: 0.711

Mathematical Citation Quotient (MCQ) 2017: 0.23

Online
ISSN
1572-9176
See all formats and pricing
More options …

# Inequalities for nonuniform wavelet frames

Firdous A. Shah
Published Online: 2019-05-16 | DOI: https://doi.org/10.1515/gmj-2019-2026

## Abstract

Gabardo and Nashed studied nonuniform wavelets by using the theory of spectral pairs for which the translation set $\mathrm{\Lambda }=\left\{0,r/N\right\}+2ℤ$ is no longer a discrete subgroup of $ℝ$ but a spectrum associated with a certain one-dimensional spectral pair. In this paper, we establish three sufficient conditions for the nonuniform wavelet system $\left\{{\psi }_{j,\lambda }\left(x\right)={\left(2N\right)}^{j/2}\psi \left({\left(2N\right)}^{j}x-\lambda \right),j\in ℤ,\lambda \in \mathrm{\Lambda }\right\}$ to be a frame for ${L}^{2}\left(ℝ\right)$. The proposed inequalities are stated in terms of Fourier transforms and hold without any decay assumptions on the generator of such a system.

MSC 2010: 42C15; 42C40; 65T60; 42A38; 42A55

## References

• [1]

O. Christensen, An Introduction to Frames and Riesz Bases, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, 2003. Google Scholar

• [2]

C. K. Chui and X. L. Shi, Inequalities of Littlewood–Paley type for frames and wavelets, SIAM J. Math. Anal. 24 (1993), no. 1, 263–277.

• [3]

I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math. 61, Society for Industrial and Applied Mathematics, Philadelphia, 1992. Google Scholar

• [4]

I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), no. 5, 1271–1283.

• [5]

L. Debnath and F. A. Shah, Lecture Notes on Wavelet Transforms, Compact Textb. Math., Birkhäuser, Cham, 2017. Google Scholar

• [6]

R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366.

• [7]

B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal. 16 (1974), 101–121.

• [8]

J.-P. Gabardo and M. Z. Nashed, An analogue of Cohen’s condition for nonuniform multiresolution analyses, Wavelets, Multiwavelets, and Their Applications (San Diego 1997), Contemp. Math. 216, American Mathematical Society, Providence (1998), 41–61. Google Scholar

• [9]

J.-P. Gabardo and M. Z. Nashed, Nonuniform multiresolution analyses and spectral pairs, J. Funct. Anal. 158 (1998), no. 1, 209–241.

• [10]

D. Li and X. Shi, A sufficient condition for affine frames with matrix dilation, Anal. Theory Appl. 25 (2009), no. 2, 166–174.

• [11]

D. Li, G. Wu and X. Yang, Unified conditions for wavelet frames, Georgian Math. J. 18 (2011), no. 4, 761–776.

• [12]

F. A. Shah and Abdullah, Nonuniform multiresolution analysis on local fields of positive characteristic, Complex Anal. Oper. Theory 9 (2015), no. 7, 1589–1608.

• [13]

F. A. Shah and M. Y. Bhat, Vector-valued nonuniform multiresolution analysis on local fields, Int. J. Wavelets Multiresolut. Inf. Process. 13 (2015), no. 4, Article ID 1550029. Google Scholar

• [14]

F. A. Shah and M. Y. Bhat, Nonuniform wavelet packets on local fields of positive characteristic, Filomat 31 (2017), no. 6, 1491–1505.

• [15]

F. A. Shah and L. Debnath, Dyadic wavelet frames on a half-line using the Walsh–Fourier transform, Integral Transforms Spec. Funct. 22 (2011), no. 7, 477–486.

• [16]

V. Sharma and P. Manchanda, Nonuniform wavelet frames in ${L}^{2}\left(ℝ\right)$, Asian-Eur. J. Math. 8 (2015), no. 2, Article ID 1550034. Google Scholar

• [17]

L. Zang and W. Sun, Inequalities for wavelet frames, Numer. Funct. Anal. Optim. 31 (2010), no. 7–9, 1090–1101.

• [18]

Z. Zhao and W. Sun, Sufficient conditions for irregular wavelet frames, Numer. Funct. Anal. Optim. 29 (2008), no. 11–12, 1394–1407.

Revised: 2018-01-10

Accepted: 2018-01-15

Published Online: 2019-05-16

Citation Information: Georgian Mathematical Journal, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X,

Export Citation

© 2019 Walter de Gruyter GmbH, Berlin/Boston.