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: 77


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.


Download data is not yet available.


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