##### Title: p-Frame Multiresolution Analysis Related to the Walsh Functions

##### Pages: 1-15

##### Cite as:

F.A. Shah, p-Frame Multiresolution Analysis Related to the Walsh Functions, Int. J. Anal. Appl., 7 (1) (2015), 1-15.#### Abstract

A generalization of the notion of p-multiresoltion analysis on a half-line, based on the theory of shift-invariant spaces is considered. In contrast to the standard setting, the associated subspace V0 of L2 (R+) has a frame, a collection of translates of the scaling function ϕ of the form {ϕ(· k)}k∈Z+ , where Z+ is the set of non-negative integers. We investigate certain properties of multiresolution subspaces which provides the quantitative criteria for the construction of p-frame multiresoltion analysis (p-FMRA) on positive half-line R+. Finally, we establish a complete characterization of all p-wavelet frames associated with p-FMRA on positive half-line R+ using the shift-invariant space theory.

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#### References

- J. J. Benedetto and S. Li, The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal., 5 (1998), 398-427.
- J. J. Benedetto and O. M. Treiber, Wavelet frames: multiresolution analysis and extension principle, in Wavelet Transforms and Time-Frequency Signal Analysis, L. Debnath, Editor, Birkhaiiser, Boston, (2000), 3-36.
- I. Daubechies, B. Han, A. Ron, and Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal., 14 (2003), 1-46.
- Yu. A. Farkov, On wavelets related to Walsh series, J. Approx. Theory, 161 (2009), 259-279.
- Yu. A. Farkov, Yu. A. Maksimov and S. A Stoganov, On biorthogonal wavelets related to the Walsh functions, Int. J. Wavelets Multiresolut. Inf. Process., 9(3) (2011), 485-499.
- Yu. A. Farkov, and E. A. Rodionov, Nonstationary wavelets related to the Walsh functions, Amer. J. Comput. Math., 2 (2012) 82-87.
- B. I. Golubov, A. V. Efimov and V. A. Skvortsov, Walsh Series and Transforms: Theory and Applications, Kluwer, Dordrecht, 1991.
- H. O. Kim and J. K. Lim, On frame wavelets associated frame multiresolution analysis, Appl. Comput. Harmon. Anal., 10 (2001) 61-70.
- S. Mallat, Multiresolution approximations and wavelets. Trans. Amer. Math. Soc., 315 (1989), 69-87.
- Meenakshi, P. Manchanda and A. H. Siddiqi, Wavelets associated with nonuniform multiresolution analysis on positive half-line, Int. J. Wavelets Multiresolut. Inf. Process., 10(2) (2012), 1250018.
- A. Ron and Z. Shen, Affine systems in L2 (Rd): the analysis of the analysis operator, J. Funct. Anal., 148 (1997), 408-447.
- A. Ron and Z. Shen, Frames and stable bases for shift-invariant subspaces of L2 (Rd), Canad. J. Math., 47 (1995), 1051-1094.
- F. Schipp, W. R. Wade and P. Simon, Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol and 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, Non-orthogonal p-wavelet packets on a half-line, Anal. Theory Appl., 28(4) (2012), 385-396.
- F. A. Shah, Biorthogonal wavelet packets related to the Walsh polynomials, J. Classical Anal., 1 (2012), 135-146.
- F. A. Shah, On some properties of p-wavelet packets via the Walsh-Fourier transform, J. Nonlinear Anal. Optimiz., 3 (2012), 185-193.
- F. A. Shah, Tight wavelet frames generated by the Walsh polynomials, Int. J. Wavelets, Multiresolut. Inf. Process., 11(6) (2013), 1350042.
- F. A. Shah and L. Debnath, Dyadic wavelet frames on a half-line using the Walsh-Fourier transform, Integ. Transf. Special Funct., 22(7) (2011), 477-486.
- P. Wojtaszczyk, A Mathematical Introduction to Wavelets, Cambridge University Press, 1997.
- Y. Xiaojiang, Semiorthogonal multiresolution analysis frames in higher dimensions, Acta Appl. Math., 111 (2010), 257-286.
- Z. H. Zhang, A characterization of generalized frame MRA’s deriving orthonormal wavelets, Acta Math. Sinica, English Series, 22(4) (2006), 1251-1260.