NECESSARY AND SUFFICIENT CONDITIONS FOR THE WAVE PACKET FRAMES ON POSITIVE HALF-LINE

Year 2016, Volume: 6 Issue: 2, 251 - 263, 01.12.2016

Abdullah Abdullah

Abstract

Abstract. In this paper, we consider wave packet systems as special cases of generalized shift-invariant systems, a concept studied by Hern´andez, Lebate and Weiss. The objective of the paper is to construct wave packet frames on positive half line. We establish necessary and sufficient conditions for the wave packet frames on positive half-line using Walsh-Fourier transform

Keywords

frame, wave packet system, sufficient condition and necessary condition.

NECESSARY AND SUFFICIENT CONDITIONS FOR THE WAVE PACKET FRAMES ON POSITIVE HALF-LINE

Abstract

References

• nces [1] Abdullah, (2013), Tight wave packet frames for L 2 (R) and H2 (R), Arab J. Math. Sci., 19(2), pp. 151-158.
• [2] Christensen,O. and Rahimi,A., (2008), An introduction to wave packet systems in L 2 (R), Indian J. Ind. Appl. Math., 1, pp. 42-57.
• [3] Christensen,O., (2003), An Introduction to Frames and Riesz Bases, Birkhuser, Boston.
• [4] Chui,C.K. and Shi,X., (1993), Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal., 24 no. 1, pp. 263-277.
• [5] C´ordoba,A. and Fefferman,C., (1978), Wave packets and Fourier integral operators, Comm. Partial Diff. Eq., 3, pp. 979
• [6] Czaja,W., Kutyniok,C. and Speegle,D., (2006), The Geometry of sets of parameters of wave packet frames, Appl. Comput. Harmon. Anal., 20, pp. 108-125.
• [7] Daubechies,I., Grossmann,A. and Meyer,Y., (1986), Painless non-orthogonal expansions, J. Math. Phys. 27 (5), pp. 1271-1283.
• [8] Duffin,R.J. and Shaeffer,A.C., (1952), A class of nonharmonic Fourier series, Trans. Am. Math. Soc. 72, pp. 341-366.
• [9] Farkov,Y.A., (2005), Orthogonal p-wavelets on R+, in Proceedings of International Conference Wavelets and Splines, St. Petersberg State University, St. Petersberg, pp. 426.
• [10] Farkov,Y.A., Maksimov,A.Y. and Stroganov,S.A., (2011), On biorthogonal wavelets related to the Walsh functions, Int. J. Wavelets, Multiresolut. Inf. Process. 9(3), pp. 485-499.
• [11] Grchenig,K., (2001), Foundations of TimeFrequency Analysis, Birkhuser, Boston.
• [12] Hern´andez,E., Labate,D. and Weiss,G., (2002), A unified characterization of reproducing systems generated by a finite family, J. Geom. Anal. 12 (4), pp. 615-662.
• [13] Hern´andez,E., Labate,D., Weiss,G. and Wilson,E., (2003), Oversampling, quasi affine frames and wave packets, Appl. Comput. Harmon. Anal. 16, pp. 111-147.
• [14] Hern´andez,E., Weiss,G. and Wilson,E., (2004), An approach to the study of wave packet systems, Contemp. Mathematics, Wavelets, Frames and Operator Theory, 345, pp. 215-235.
• [15] Meenakshi, Manchanda,P. and Siddiqi,A.H., (2011), Wavelets associated with Nonuniform multiresolution analysis on positive half line, Int. J. Wavelets, Multiresolut. Inf. Process., 10(2), 1250018, pp. 27.
• [16] Ron,A. and Shen,Z., (2005), Generalized shift-invariant systems, Const. Appr., 22, pp. 1-45.
• [17] Shah,F.A. and Abdullah, (2012), Necessary condition for the existence of wave packet frames, Southeast Asian Bull. of Math., 36, pp. 287-292.
• [18] Shah,F.A., (2012), Gabor frames on a half-line, J. cont. Math. Anal. 5, 47 (2012), pp. 251-260.
• [19] Shah,F.A., (2009), Construction of wavelet packets on p-adic field, Int. J. Wavelets, Multiresolut. Inf. Process., 7, pp. 553-565.
• [20] Shah,F.A., (2013), Tight wavelet frames generated by the Walsh polynomials, Int. J. Wavelets, Multiresolut. Inf. Process., 11(6), 1350042, pp. 15.
• [21] Shah,F.A. and Debnath,L., (2011), Dyadic wavelet frames on a half-line using the Walsh-Fourier transform, Integ. Trans. Special Funct., 22(7), pp. 477-486.
• [22] Shah,F.A. and Debnath,L., (2011), p-Wavelet frame packets on a half-line using the Walsh–Fourier transform, Integ. Trans. Special Funct., 22, pp. 907-917.