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Inequalities for nonuniform wavelet frames

  • Firdous A. Shah ORCID logo EMAIL logo
Published/Copyright: May 16, 2019
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Abstract

Gabardo and Nashed studied nonuniform wavelets by using the theory of spectral pairs for which the translation set Λ={0,r/N}+2 is no longer a discrete subgroup of but a spectrum associated with a certain one-dimensional spectral pair. In this paper, we establish three sufficient conditions for the nonuniform wavelet system {ψj,λ(x)=(2N)j/2ψ((2N)jx-λ),j,λΛ} to be a frame for L2(). The proposed inequalities are stated in terms of Fourier transforms and hold without any decay assumptions on the generator of such a system.

Acknowledgements

The author is deeply indebted to the referees for their constructive comments and suggestions which significantly improved the presentation of the paper.

References

[1] O. Christensen, An Introduction to Frames and Riesz Bases, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, 2003. 10.1007/978-0-8176-8224-8Search in 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. 10.1137/0524017Search in Google Scholar

[3] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math. 61, Society for Industrial and Applied Mathematics, Philadelphia, 1992. 10.1137/1.9781611970104Search in Google Scholar

[4] I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), no. 5, 1271–1283. 10.1515/9781400827268.372Search in Google Scholar

[5] L. Debnath and F. A. Shah, Lecture Notes on Wavelet Transforms, Compact Textb. Math., Birkhäuser, Cham, 2017. 10.1007/978-3-319-59433-0Search in Google Scholar

[6] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366. 10.1090/S0002-9947-1952-0047179-6Search in Google Scholar

[7] B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal. 16 (1974), 101–121. 10.1016/0022-1236(74)90072-XSearch in Google Scholar

[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. 10.1090/conm/216/02963Search in 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. 10.1006/jfan.1998.3253Search in Google Scholar

[10] D. Li and X. Shi, A sufficient condition for affine frames with matrix dilation, Anal. Theory Appl. 25 (2009), no. 2, 166–174. 10.1007/s10496-009-0166-0Search in Google Scholar

[11] D. Li, G. Wu and X. Yang, Unified conditions for wavelet frames, Georgian Math. J. 18 (2011), no. 4, 761–776. 10.1515/GMJ.2011.0047Search in Google Scholar

[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. 10.1007/s11785-014-0412-0Search in Google Scholar

[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. 10.1142/S0219691315500290Search in 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. 10.2298/FIL1706491SSearch in Google Scholar

[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. 10.1080/10652469.2010.520528Search in Google Scholar

[16] V. Sharma and P. Manchanda, Nonuniform wavelet frames in L2(), Asian-Eur. J. Math. 8 (2015), no. 2, Article ID 1550034. 10.1142/S1793557115500345Search in Google Scholar

[17] L. Zang and W. Sun, Inequalities for wavelet frames, Numer. Funct. Anal. Optim. 31 (2010), no. 7–9, 1090–1101. 10.1080/01630563.2010.512552Search in Google Scholar

[18] Z. Zhao and W. Sun, Sufficient conditions for irregular wavelet frames, Numer. Funct. Anal. Optim. 29 (2008), no. 11–12, 1394–1407. 10.1080/01630560802594621Search in Google Scholar

Received: 2017-04-17
Revised: 2018-01-10
Accepted: 2018-01-15
Published Online: 2019-05-16
Published in Print: 2021-02-01

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