The necessary and sufficient conditions for wavelet frames in Sobolev space over local fields
DOI:
https://doi.org/10.5269/bspm.40871Abstract
In this paper we construct wavelet frame on Sobolev space. A necessary condition and suffcient conditions for wavelet frames in Sobolev space are given.
References
1. C. K. Chui, An Introduction to Wavelets, Wavelet Analysis and Its Applications, Academic Press, Boston, MA, 1, (1992).
2. S. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2(R), Trans. Amer. Math. Soc. 315, 69-87, (1989).
3. Y. Meyer, Wavelets and Operators, Cambridge University Press, Cambridge, 1992.
4. S. Dahlke, Multiresolution analysis and wavelets on locally compact abelian groups. in Wavelets, images, and surface fitting, A K Peters, 141-156, (1994).
5. J. J. Benedetto and R. L. Benedetto, A wavelet theory for local fields and related groups, J. Geom. Anal. 14, 423-456, (2004).
6. R. L. Benedetto, Examples of wavelets for local fields, in: Wavelets, Frames and Operator Theory, Contemporary Mathematics, American Mathematical Society, Providence, RI,345, 27-47,(2004).
7. S. Albeverio and S. Kozyrev, Multidimensional basis of p-adic wavelets and representation theory, P-Adic Numbers Ultrametric, Anal. Appl. 1, 181-189, (2009).
8. A. Y. Khrennikov, V. M. Shelkovich and M. Skopina, p-adic refinable functions and MRA-based wavelets, J. Approx. Theory 161, 226-238 (2009).
9. S. Kozyrev, Wavelet theory as p-adic spectral analysis Izv. Ross. Akad. Nauk Ser. Mat. 66, 149-158, (2002).
10. H. Jiang, D. Li and N. Jin, Multiresolution analysis on local fields, J. Math. Anal. Appl. 294, 523-532, (2004).
11. D. Ramakrishnan and R. J. Valenza, Fourier Analysis on Number Fields, Graduate Texts in Mathematics, Springer-Verlag, New York,186 (1999).
12. M. H. Taibleson, Fourier Analysis on Local Fields, Mathematical Notes, Princeton University Press, Princeton, NJ,15 (1975).
13. Biswaranjan Behera and Qaiser Jahan, Multiresolution analysis on local fields and characterization of scaling functions, Adv. Pure Appl. Math. 3 , 181-202, (2012).
14. F. Bastin and P.Laubin, Regular Compactly Supported Wavelets in Sobolev spaces, Duke Mathematical Journal, 87 (3), 481-508, (1997).
15. Ashish Pathak and Guru P. Singh, Wavelet in Sobolev space over local fields of positive characteristic, Int. J. of Wavelets Multiresolunt. Inf. Process, 16(3), 16pp, (2018).
16. Ashish Pathak, Dileep Kumar and Guru P. Singh Multiresolution Analysis on Sobolev space over local fields of positive characteristic and Characterization of scaling function (preprint).
2. S. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2(R), Trans. Amer. Math. Soc. 315, 69-87, (1989).
3. Y. Meyer, Wavelets and Operators, Cambridge University Press, Cambridge, 1992.
4. S. Dahlke, Multiresolution analysis and wavelets on locally compact abelian groups. in Wavelets, images, and surface fitting, A K Peters, 141-156, (1994).
5. J. J. Benedetto and R. L. Benedetto, A wavelet theory for local fields and related groups, J. Geom. Anal. 14, 423-456, (2004).
6. R. L. Benedetto, Examples of wavelets for local fields, in: Wavelets, Frames and Operator Theory, Contemporary Mathematics, American Mathematical Society, Providence, RI,345, 27-47,(2004).
7. S. Albeverio and S. Kozyrev, Multidimensional basis of p-adic wavelets and representation theory, P-Adic Numbers Ultrametric, Anal. Appl. 1, 181-189, (2009).
8. A. Y. Khrennikov, V. M. Shelkovich and M. Skopina, p-adic refinable functions and MRA-based wavelets, J. Approx. Theory 161, 226-238 (2009).
9. S. Kozyrev, Wavelet theory as p-adic spectral analysis Izv. Ross. Akad. Nauk Ser. Mat. 66, 149-158, (2002).
10. H. Jiang, D. Li and N. Jin, Multiresolution analysis on local fields, J. Math. Anal. Appl. 294, 523-532, (2004).
11. D. Ramakrishnan and R. J. Valenza, Fourier Analysis on Number Fields, Graduate Texts in Mathematics, Springer-Verlag, New York,186 (1999).
12. M. H. Taibleson, Fourier Analysis on Local Fields, Mathematical Notes, Princeton University Press, Princeton, NJ,15 (1975).
13. Biswaranjan Behera and Qaiser Jahan, Multiresolution analysis on local fields and characterization of scaling functions, Adv. Pure Appl. Math. 3 , 181-202, (2012).
14. F. Bastin and P.Laubin, Regular Compactly Supported Wavelets in Sobolev spaces, Duke Mathematical Journal, 87 (3), 481-508, (1997).
15. Ashish Pathak and Guru P. Singh, Wavelet in Sobolev space over local fields of positive characteristic, Int. J. of Wavelets Multiresolunt. Inf. Process, 16(3), 16pp, (2018).
16. Ashish Pathak, Dileep Kumar and Guru P. Singh Multiresolution Analysis on Sobolev space over local fields of positive characteristic and Characterization of scaling function (preprint).
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Published
2020-10-09
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Funding data
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Council of Scientific and Industrial Research, India
Grant numbers 09/013(0647)/2016 - EMR - 1
