A Characterization of Orthonormal Multilevel Wavelet Families in Sobolev Space over Local Fields of Positive Characteristic

Article Sidebar

PDF
Published: Aug 1, 2021
DOI: https://doi.org/10.5556/j.tkjm.52.2021.3327
Keywords:
Wavelets Orthonormal wavelet families Local fields Wavelets ; Orthonormal wavelet families; Local elds; Sobolev space; Fourier transform; Multiwavelet; orthonormal; Fourier transform; local field.

Main Article Content

Ashish Pathak
Banaras Hindu University
https://orcid.org/0000-0001-7581-3872
Dileep Kumar
Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-221005, India.

Abstract




In this paper, a characterization of orthonormal multilevel wavelet families in Sobolev space over a local fields of positive characteristic $(H^s(K))$ is established. Finally an example is presented.




Article Details

How to Cite
Pathak, A., & Kumar, D. (2021). A Characterization of Orthonormal Multilevel Wavelet Families in Sobolev Space over Local Fields of Positive Characteristic. Tamkang Journal of Mathematics, 52(3), 349–361. https://doi.org/10.5556/j.tkjm.52.2021.3327
Issue
Vol. 52 No. 3 (2021)
Section
Papers
Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

References

A. Pathak, Continuous Wavelet Transformon Local Fields, Bol. Soc. Paran. Mat. 34(2016), 113-121.

A. Pathak and D. Kumar, Characterization of Multiwavelets and MRA Wavelets in Hs(F), International Journal of Applied and Computational Mathematics 5(6) (2019), 143.

A. Pathak and D. Kumar, Existence of unconditional wavelet basis of Lp−norm over a local fields of positive characteristic (communicated).

A. Pathak, D. Kumar and G. P. Singh, The Necessary Condition and Sufficient conditions for Wavelet Frames in Sobolev Space over Local field of Positive characteristic, Bol. Soc. Paran. Mat., 39(3)(2021) 81-92.

A. Pathak and G. P Singh , Wavelet in Sobolve space over local fields of positive character- iscic, Int. J. of Wavelets Multiresolunt. Inf. Process 3 (2018), 16.

A. Pathak and G. P. Singh, Biorthogonal Wavelets in Sobolev Space Over Local Fields of Positive Characteristic. International Journal of Applied and Computational Mathematics 6(2) (2020), 25.

A. Pathak and G. P. Singh, Multilevel Wavelet packets in Sobolev space over local fields of positive characteristic (communicated).

A. Ron and Z. Shen, Frames and stable bases for shift invariant subspaces of L2(Rd). Can. J. Math. 47(1995), 1051-1094.

A. Ron and Z. Shen, Affine systems in L2(Rd) the analysis of the analysis operator. J. Funct. Anal. 148 (1997), 408-447.

B. Behera, Haar wavelets on the Lebesgue spaces of local fields of positive characteristic, Colloquium Mathematicum. 2 (2014), 149-168.

B. Behera and Q. Jahan, Multiresolution analysis on local fields and characterization of scaling functions, Adv. Pure Appl. Math. 3 (2012), 181-202.

B. Behera and Q. Jahan, Characterization of wavelets and MRA wavelets on local fields of positive characteristic, Collect. Math. 66 (2015), 33-53.

C. K. Chui, An Introduction to Wavelets, Wavelet Analysis and Its Applications 1, Academic Press, Boston, MA, (1992).

C. Chui, X. Shi and J. Stocker, Affine frames, quasi-affine frames, and their duals. Adv. Comput. Math. 8 (1998), 1-17.

D. F. Walnut, An introduction to wavelet analysis, Birkhauser (2002).

E. Hernández and G. Weiss, A First Course on Wavelets, CRC Press, Boca Raton, FL, (1996).

G. Gripenberg, A necessary and sufficient condition for the existance of a father wavelet. Stud. Math.114 (1995), 207-226.

H. Jiang, D. Li and N. Jin, Multiresolution analysis on local fields, J. Math. Anal. Appl. 294 (2004), 523-532.

I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 61 (1992).

J. J. Benedetto and R. L. Benedetto, A wavelet theory for local fields and related groups, J. Geom. Anal. 14 (2004), 423-456.

M. Bownik, On characterization of multiwavelets in L2(Rn).Proc. Am. Math. Soc. 129 (2001), 3265-3274.

M. Bownik, The structure of shift invariant subspaces of L2(Rn). J. Funct. Anal. 177(2000), 282-309.

M. H. Taibleson, Fourier Analysis on Local Fields, Mathematical Notes, Princeton University Press, Princeton, NJ 15 (1975).

R. L. Benedetto, Examples of wavelets for local fields, in: Wavelets, Frames and Operator Theory, Contemporary Mathematics, American Mathematical Society, Providence, RI 345 (2004), 27-47.

S. Dahlke, Multiresolution analysis and wavelets on locally compact abelian groups. in: Wavelets, images, and surface fitting, A K Peters (1994), 141-156pp.

S. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2(R), Trans. Amer. Math. Soc. 315 (1989), 69-87.

X. Wang, The study of wavelets from the properties of their Fourier transform, PhD. Thesis, Washington University (1995).