Wavelet bi-frames on local fields


  • Owais Ahmad National Institute of Technology, Hazratbal, Srinagar, Jammu and Kashmir, India https://orcid.org/0000-0002-9903-2656
  • Neyaz Ahmad National Institute of Technology, Hazratbal, Srinagar, Jammu and Kashmir, India
  • Mobin Ahmad Jazan University, Jazan, Saudi Arabia https://orcid.org/0000-0002-4131-3391




Periodic wavelet frame, Bi-frame, Local field, Fourier transform
Abstract views: 166


In this paper, we introduce the notion of periodic wavelet bi-frames on local fields and establish the theory for the construction of periodic Bessel sequences and periodic wavelet bi-frames on local fields.


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O. Ahmad, M.Y. Bhat, N. A. Sheikh, Construction of Parseval Framelets Associated with GMRA on Local Fields of Positive Characteristic, Numerical Functional Analysis and Optimization (2021), https://doi.org/10.1080/01630563.2021.1878370. DOI: https://doi.org/10.1080/01630563.2021.1878370

O. Ahmad, N. Ahmad, Construction of Nonuniform Wavelet Frames on Non-Archimedean Fields, Math. Phy. Anal. and Geometry, 23 (47) (2020)., https://doi.org/10.1007/s11040-020-09371-1 DOI: https://doi.org/10.1007/s11040-020-09371-1

O. Ahmad, N. A Sheikh, K. S Nisar, F. A. Shah, Biorthogonal Wavelets on Spectrum, Math. Methods in Appl. Sci, (2021) pp. 1–12, https://doi.org/10.1002/mma.7046. DOI: https://doi.org/10.22541/au.160075814.43954669

O. Ahmad, Nonuniform Periodic Wavelet Frames on Non-Archimedean Fields, Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica, pp. 1-17, (2) (2020), https://doi.org/10.17951/a.2020.74.2.1-17. DOI: https://doi.org/10.17951/a.2020.74.2.1-17

O. Ahmad, N. A Sheikh, Explicit Construction of Tight Nonuniform Framelet Packets on Local Fields, Operators and Matrices 15 (1) (2021), 131–149, dx.doi.org/10.7153/oam-2021-15-10 DOI: https://doi.org/10.7153/oam-2021-15-10

O. Ahmad, N.A. Sheikh, M. A. Ali, Nonuniform nonhomogeneous dual wavelet frames in Sobolev spaces in $L^2(mathbb K)$, Afrika Math., (2020), https://doi.org/10.1007/s13370-020-00786-1. DOI: https://doi.org/10.1007/s13370-020-00786-1

O. Ahmad and N. A. Sheikh, On Characterization of nonuniform tight wavelet frames on local fields, Anal. Theory Appl., 34 (2018) 135-146, https://doi.org/10.4208/ata.2018.v34.n2.4 DOI: https://doi.org/10.4208/ata.2018.v34.n2.4

O. Ahmad, F. A. Shah and N. A. Sheikh, Gabor frames on non-Archimedean fields, International Journal of Geometric Methods in Modern Physics, 15 (2018) 1850079 (17 pages), https://doi.org/10.1142/S0219887818500792 DOI: https://doi.org/10.1142/S0219887818500792

S. Albeverio, S. Evdokimov, and M. Skopina, $p$-adic nonorthogonal wavelet bases, Proc. Steklov Inst. Math., 265 (2009) 1-12. https://doi.org/10.1134/S0081543809020011 DOI: https://doi.org/10.1134/S0081543809020011

S. Albeverio, S. Evdokimov, and M. Skopina, $p$-adic multiresolution analysis and wavelet frames, Fourier Anal. Appl., 16 (2010) 693-714, https://doi.org/10.1007/s00041-009-9118-5 DOI: https://doi.org/10.1007/s00041-009-9118-5

S. Albeverio, A. Khrennikov, and V. Shelkovich, Theory of p-adic Distributions: Linear and Nonlinear Models, Cambridge University Press, 2010. DOI: https://doi.org/10.1017/CBO9781139107167

J. J. Benedetto and R. L. Benedetto, A wavelet theory for local fields and related groups, J. Geom. Anal., 14 (2004) 423-456, https://doi.org/10.1007/BF02922099 DOI: https://doi.org/10.1007/BF02922099

O.~ Christensen and S. ~ S.~ Goh, The unitary extension principle on locally compact abelian groups, Appl. Comput. Harmon.Anal., 47 (1) (2019) 1-29., https://doi.org/10.1016/j.acha.2017.07.004. DOI: https://doi.org/10.1016/j.acha.2017.07.004

R. J. Duffin and A. C. Shaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952) 341-366, https://doi.org/10.2307/1990760 DOI: https://doi.org/10.1090/S0002-9947-1952-0047179-6

S. Evdokimov and M. Skopina, 2-adic wavelet bases, Proc. Steklov Inst. Math., 266 (2009), S143-S154, https://doi.org/10.1134/S008154380906011X DOI: https://doi.org/10.1134/S008154380906011X

Y. Farkov, Orthogonal wavelets on locally compact abelian groups, Funct. Anal. Appl., 31 (1997), 294-296, https://doi.org/10.1007/BF02466067 DOI: https://doi.org/10.1007/BF02466067

Y. Farkov, Multiresolution Analysis and Wavelets on Vilenkin Groups, Facta Universitatis (NIS), Ser.: Elec. Energ., 21 (2008) 309-325, https://doi.org/10.2298/FUEE0803309F DOI: https://doi.org/10.2298/FUEE0803309F

H.K. Jiang, D.F. Li and N. Jin, Multiresolution analysis on local fields, J. Math. Anal. Appl. 294 (2004) 523-532. https://doi.org/10.1016/j.jmaa.2004.02.026 DOI: https://doi.org/10.1016/j.jmaa.2004.02.026

A.Khrennikov andV.Shelkovich, Non-Haar $p$-adic wavelets and their application to pseudo-differential operators and equations, Appl. Comput. Harmon. Anal., 28 (2010) 1-23, https://doi.org/10.1016/j.acha.2009.05.007 DOI: https://doi.org/10.1016/j.acha.2009.05.007

A. Khrennikov, V. Shelkovich, and M. Skopina, $p$-adic refinable functions and MRA-based wavelets, J. Approx. Theory. 161 (2009) 226-238, https://doi.org/10.1016/j.jat.2008.08.008 DOI: https://doi.org/10.1016/j.jat.2008.08.008

S. Kozyrev and A. Khrennikov, $p$-adic integral operators in wavelet bases, Doklady Math., 83 (2011) 209–212, https://doi.org/10.1134/S1064562411020220 DOI: https://doi.org/10.1134/S1064562411020220

S. Kozyrev, A. Khrennikov, and V. Shelkovich, $p$-Adic wavelets and their applications, Proc. Steklov Inst. Math., 285 (2014) 157-196, https://doi.org/10.1134/S0371968514020125 DOI: https://doi.org/10.1134/S0081543814040129

W. C. Lang, Orthogonal wavelets on the Cantor dyadic group, SIAM J. Math. Anal., 27 (1996) 305–312, https://doi.org/10.1137/S0036141093248049 DOI: https://doi.org/10.1137/S0036141093248049

W. C. Lang, Wavelet analysis on the Cantor dyadic group, Houston J. Math., 24 (1998) 533-544.

W. C. Lang, Fractal multiwavelets related to the cantor dyadic group, Int. J. Math. Math. Sci., 21 (1998) 307-314., https://doi.org/10.1155/S0161171298000428 DOI: https://doi.org/10.1155/S0161171298000428

D. F. Li and H. K. Jiang, The necessary condition and sufficient conditions for wavelet frame on local fields, J. Math. Anal. Appl., 345 (2008) 500-510, https://doi.org/10.1016/j.jmaa.2008.04.031 DOI: https://doi.org/10.1016/j.jmaa.2008.04.031

S. G. Mallat, Multiresolution approximations and wavelet orthonormal bases of $L^2(mathbb R)$, Trans. Amer. Math. Soc., 315 (1989) 69-87, https://doi.org/10.1090/S0002-9947-1989-1008470-5 DOI: https://doi.org/10.1090/S0002-9947-1989-1008470-5

A.~ Ron and Z.~ Shen, Affine systems in $L^2(mathbb{R}^d)$: the analysis of the analysis operator, J. Funct. Anal., 148 (1997) 408 -447, https://doi.org/10.1006/jfan.1996.3079 DOI: https://doi.org/10.1006/jfan.1996.3079

F.A. Shah and O. Ahmad, Wave packet systems on local fields, Journal of Geometry and Physics, 120 (2017) 5-18, https://doi.org/10.1016/j.geomphys.2017.05.015 DOI: https://doi.org/10.1016/j.geomphys.2017.05.015

F. A. Shah, O. Ahmad and A. Rahimi, Frames Associated with Shift Invariant Spaces on Local Fields, Filomat 32 (9) (2018) 3097-3110, https://doi.org/10.2298/FIL1809097S DOI: https://doi.org/10.2298/FIL1809097S

F. A. Shah, O. Ahmad and N. A. Sheikh, Orthogonal Gabor systems on local fields. Filomat 31 (16) (2017) 5193-5201, https://doi.org/10.2298/FIL1716193S DOI: https://doi.org/10.2298/FIL1716193S

F. A. Shah, O. Ahmad and N. A. Sheikh, Some new inequalities for wavelet frames on local fields. Anal. Theory Appl., 33 (2) (2017) 134-148, https://doi.org/10.4208/ata.2017.v33.n2.4 DOI: https://doi.org/10.4208/ata.2017.v33.n2.4

M. H. Taibleson, Fourier Analysis on Local Fields, Princeton University Press, Princeton, NJ, 1975.

Y.Z. Li, H. F. Jia, The construction of multivariate periodic wavelet bi-frames, J. Math. Anal. Appl. 412 (2014) 852-865, https://doi.org/10.1016/j.jmaa.2013.11.021 DOI: https://doi.org/10.1016/j.jmaa.2013.11.021




How to Cite

Ahmad, O., Ahmad, N., & Ahmad, M. . (2022). Wavelet bi-frames on local fields. J. Numer. Anal. Approx. Theory, 51(2), 124–143. https://doi.org/10.33993/jnaat512-1265