ON THE CHARACTERIZATION OF SCALING FUNCTIONS ON NON-ARCHEMEDEAN FIELDS

Ishtaq Ahmed     (National Institute of Technology, Jammu and Kashmir, Srinagar-190006, India)
Owias Ahmad     (National Institute of Technology, Jammu and Kashmir, Srinagar-190006, India)
Neyaz Ahmad Sheikh     (National Institute of Technology, Jammu and Kashmir, Srinagar-190006, India)

Abstract


In real life application all signals are not obtained from uniform shifts; so there is a natural question regarding analysis and decompositions of these types of signals by a stable mathematical tool.  This gap was filled by Gabardo and Nashed [11]   by establishing a constructive algorithm based on the theory of spectral pairs for constructing non-uniform wavelet basis in \(L^2(\mathbb R)\). In this setting, the associated translation set \(\Lambda =\left\{ 0,r/N\right\}+2\,\mathbb Z\) is no longer a discrete subgroup of \(\mathbb R\) but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we characterize the scaling function for non-uniform multiresolution analysis on local fields of positive characteristic (LFPC). Some properties of wavelet scaling function associated with non-uniform multiresolution analysis (NUMRA) on LFPC are also established.


Keywords


Scaling function, Fourier transform, Local field, NUMRA

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References


  1. Ahmad I., Sheikh N.A. \(a\)-inner product on local fields of positive characteristic. J. Nonlinear Anal. Appl., 2018. Vol. 2018, No. 2. P. 64–75.
  2. Ahmad I., Sheikh N.A. Dual wavelet frames in Sobolev spaces on local fields of positive characteristic. Filomat, 2020. Vol. 34, No. 6. P. 2091–2099. DOI: 10.2298/FIL2006091A
  3. Ahmad O., Sheikh N.A. Explicit construction of tight nonuniform framelet packets on local fields. Oper. Matrices, 2021. Vol. 15, No. 1. P. 131–149. DOI: 10.7153/oam-2021-15-10
  4. Ahmad O., Sheikh N.A. On characterization of nonuniform tight wavelet frames on local fields. Anal. Theory Appl., 2018. Vol. 34. P. 135–146. DOI: 10.4208/ata.2018.v34.n2.4
  5. Albeverio S., Evdokimov S., Skopina M. \(p\)-adic multiresolution analysis and wavelet frames. J. Fourier Anal. Appl., 2010. Vol. 16. P. 693–714. DOI: 10.1007/s00041-009-9118-5
  6. Albeverio S., Kozyrev S. Multidimensional basis of p-adic wavelets and representation theory. \(p\)-Adic Num. Ultrametric Anal. Appl., 2009. Vol. 1. No. 3. P. 181–189. DOI: 10.1134/S2070046609030017
  7. Behera B., Jahan Q. Multiresolution analysis on local fields and characterization of scaling functions. Adv. Pure Appl. Math., 2012. Vol. 3, No. 2. P. 181–202. DOI: 10.1515/apam-2011-0016
  8. Behera B., Jahan Q. Characterization of wavelets and MRA wavelets on local fields of positive characteristic. Collect. Math., 2015. Vol. 66, No. 1. P. 33–53. DOI: 10.1007/s13348-014-0116-9
  9. Benedetto J.J., Benedetto R.L. A wavelet theory for local fields and related groups. J. Geom. Anal., 2004. Vol. 14, No. 3. P. 423–456.
  10. Cifuentes P., Kazarian K.S., Antolín A.S. Characterization of scaling functions in multiresolution analysis. Proc. Am. Math. Soc., 2005. Vol. 133, No. 4. P. 1013–1023. 
  11. Gabardo J.-P., Nashed M.Z. Nonuniform multiresolution analyses and spectral pairs. J. Funct. Anal., 1998. Vol. 158, No. 1. P. 209–241. DOI: 10.1006/jfan.1998.3253
  12. Gabardo J.-P., Yu X. Wavelets associated with nonuniform multiresolution analyses and one-dimensional spectral pairs. J. Math. Anal. Appl., 2006. Vol. 323, No. 2. P. 798–817. DOI: 10.1016/j.jmaa.2005.10.077
  13. Jiang H., Li D., Jin N. Multiresolution analysis on local fields. J. Math. Anal. Appl., 2004. Vol. 294, No. 2. P. 523–532. DOI: 10.1016/j.jmaa.2004.02.026
  14. Khrennikov A.Yu., Kozyrev S.V. Wavelets on ultrametric spaces. Appl. Comput. Harmon. Anal., 2005. Vol. 19. P. 61–76. DOI: 10.1016/j.acha.2005.02.001
  15. Khrennikov A.Yu., Shelkovich V.M. An infinite family of \(p\)-adic non-Haar wavelet bases and pseudo-differential operators. \(p\)-Adic Num. Ultrametric Anal. Appl., 2009. Vol. 1. P. 204–216. DOI: 10.1134/S2070046609030030
  16. Khrennikov A.Yu., Shelkovich V.M. Skopina M. \(p\)-adic orthogonal wavelet bases. \(p\)-Adic Num. Ultrametric Anal. Appl., 2009. Vol. 1, No. 2. P. 145–156. DOI: 10.1134/S207004660902006X
  17. Khrennikov A.Yu., Shelkovich V.M. Skopina M. \(p\)-adic refinable functions and MRA-based wavelets. J. Approx. Theory, 2009. Vol. 161, No. 1. P. 226–238. DOI: 10.1016/j.jat.2008.08.008
  18. Kozyrev S.V. Wavelet theory as \(p\)-adic spectral analysis. Izv. Math., 2002. Vol. 66, No. 2. P. 149–158.
  19. Li D., Jiang H. The necessary condition and sufficient conditions for wavelet frame on local fields. J. Math. Anal. Appl., 2008. Vol. 345, No. 1. P. 500–510. DOI: 10.1016/j.jmaa.2008.04.031
  20. Madych W.R. Some elementary properties of multiresolution analysis of \(L^2(\mathbb R^n)\). In: Wavelets: A Tutorial in Theory and Applications. Vol. 2: Wavelet Analysis and Its Applications. Chui C.K. (ed.), 1992. P. 259–294. DOI: 10.1016/B978-0-12-174590-5.50015-0
  21. Mallat S.G. Multiresolution approximations and wavelet orthonormal bases of \(L^2(\mathbb R)\). Trans. Amer. Math. Soc., 1989. Vol. 315, No. 1. P. 69–87.
  22. Shah F.A., Ahmad O. Wave packet systems on local fields. J. Geom. Phys., 2017. Vol. 120. P. 5–18. DOI: 10.1016/j.geomphys.2017.05.015
  23. Shah F.A., Abdullah. Nonuniform multiresolution analysis on local fields of positive characteristic. Complex Anal. Oper. Theory, 2015. Vol. 9. P. 1589–1608. DOI: 10.1007/s11785-014-0412-0
  24. Shukla N.K., Maury S.C. Super-wavelets on local fields of positive characteristic. Math. Nachr., 2018. Vol. 291, No. 4. P. 704–719. DOI: 10.1002/mana.201500344
  25. Taibleson M.H. Fourier Analysis on Local Fields. (MN-15). Princeton, NJ: Princeton University Press, 1975. 306 p.
  26. Zhang Z. Supports of Fourier transforms of scaling functions. Appl. Comput. Harmon. Anal., 2007. Vol. 22, No. 2. P. 141–156. DOI: 10.1016/j.acha.2006.05.007




DOI: http://dx.doi.org/10.15826/umj.2021.1.001

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