$A$-transform of Wavelet Frames

F. A. Shah, N. A. Sheikh


In this paper, we introduced the concept of $A$-transform $A = (a_{p,q,j,k})$ and study the action of $A$ on $f\in L^2(\mathbb{R}^+)$ and on its wavelet coefficients. Further, we also establish the Parseval frame condition for $A$-transform of $f \in L^2(\mathbb{R}^+)$ whose wavelet series expansion is known.


$A$-transform; Wavelet frame; Walsh function; Walsh-Fourier transform

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P.G. Casazza and O. Christensen, Weyl-Heisenberg frames for subspaces of $L^2(mathbb{R})$, Proc. Amer. Math. Soc. 129 (2001), 145–154.

O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003.

C.K. Chui and X.L. Shi, Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal. 24 (1993), 263–277.

C.K. Chui and X.L. Shi, Orhtonormal wavelets and tight frames with arbitrary real dilations, Appl. Comput. Harmonic Anal. 9 (2000), 243–264.

S. Dahlke, Multiresolution analysis and wavelets on locally compact Abelian groups, in Wavelets, Images and Surface Fitting, P.J. Laurent, A. Le Mehaute and L.L. Schumaker (editors), A.K.Peters,Wellesley, (1994), 141–156.

I. Daubechies, Ten Lecture on Wavelets, CBMS-NSF Regional Conferences in Applied Mathematics (SIAM, Philadelphia, 1992).

I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27(5) (1986), 1271–1283.

R.J. Duffin and A.C. Shaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366.

Yu.A. Farkov, Orthogonal wavelets with compact support on locally compact Abelian groups, Izv. Math. 69(3) (2005), 623–650.

Yu.A. Farkov, On wavelets related toWalsh series, J. Approx. Theory, 161(1) (2009), 259–279.

B.I. Golubov, A.V. Efimov and V.A. Skvortsov, Walsh Series and Transforms: Theory and Applications, Kluwer, Dordrecht, 1991.

J. Kovacevic and A. Chebira, Life beyond bases: the advent of frames-I, IEEE Sig. Process. Magz. 24(4) (2007), 86–104.

J. Kovacevic and A. Chebira, Life beyond bases: the advent of frames-I, IEEE Sig. Process. Magz. 24(5) (2007), 115–125.

W.C. Lang, Orthogonal wavelets on the Cantor dyadic group, SIAM J. Math. Anal. 27 (1996), 305–312.

W.C. Lang, Fractal multiwavelets related to the Cantor dyadic group, Int. J. Math. Math. Sci. 21 (1998), 307–317.

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

S.G. Mallat, Multiresolution approximations and wavelet orthonormal bases of $L^2(mathbb{R})$, Trans. Amer. Math. Soc. 315 (1989), 69–87.

F. Móricz and B.E. Rhoades, Comparison theorems for double summability methods, Publ. Math. Debrecen, 36 (1989), 207–220.

V.Yu. Protasov and Yu.A. Farkov, Dyadic wavelets and refinable functions on a half-line, Sb. Math. 197(10) (2006), 1529–1558.

F. Schipp, W.R. Wade and P. Simon, Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol — New York, 1990.

F.A. Shah, Construction of wavelet packets on $p$-adic field, Int. J. Wavelets Multiresolut. Inf. Process. 7(5) (2009), 553–565.

F.A. Shah and L. Debnath, Dyadic wavelet frames on a half-line using the Walsh-Fourier transform, Integ.Transf. Spec. Funct. 22(7) (2011), 477–486.

F.A. Shah and L. Debnath, $p$-Wavelet frame packets on a half-line using the Walsh-Fourier transform, Integ. Transf. Spec. Funct. 22(12) (2011), 907–917.

N.A. Sheikh and M. Mursaleen, Infinite matrices, wavelets coefficients and frames, Int. J. Mathematics and Mathematical Sciences 67 (2004), 3695–3702.

X.L. Shi and F. Chen, Necessary condition and sufficient condition for affine frames, Science in China (Series A) 48 (2005), 1369–1378.

DOI: http://dx.doi.org/10.26713%2Fcma.v3i3.211


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