Firdous A. Shah

CONSTRUCTION OF WAVELET PACKETS ON p-ADIC FIELD

Wavelet packets are subsets of a multiresolution analysis and retain many of the orthogonality, smoothness and localization properties of their parent wavelets. In this paper, we study the construc...

p-Wavelet frame packets on a half-line using the Walsh–Fourier transform

This paper deals with a construction of wavelet frame packets on the positive half-line ℝ+ using the splitting trick for frames. It is shown that, as long as finitely many splitting steps are applied, the resulting sequence of functions is a frame of L 2(ℝ+). If the matrix M(ξ) associated with the splitting is unitary, then the splitting can be applied infinitely many times to prove the existence of frame with the frame bounds as shown in Theorem 3.3.

Tight wavelet frames on local fields

Summary In this paper, some algorithms for constructing tight wavelet frames on local fields using the unitary extension principles are suggested. We present a sufficient condition for finite number of functions to form a tight wavelet frame and establish general principles for constructing tight wavelet frames on local fields

Dyadic wavelet frames on a half-line using the Walsh–Fourier transform

This paper deals with a construction of dyadic wavelet frames on a positive half-line ℝ+ using the Walsh–Fourier transform. We prove necessary and sufficient conditions for the system to be a wavelet frame in L 2(ℝ+). These conditions are found to be better than those of Daubechies [Ten Lecture on Wavelets, NSF-CBMS Regional Conferences in Applied Mathematics, Vol. 61, SIAM, Philadelphia, PA, 1992], Chui and Shi [Inequalities of Littlewood–Paley type for frames and wavelets, SIAM J. Math. Anal. 24 (1993), pp. 263–277], Casazza and Christenson [Weyl–Heisenberg frames for subspaces of L 2(ℝ), Proc. Amer. Math. Soc. 129 (2001), pp. 145–154] and Christenson [Frames, Riesz bases, and discrete Gabor/wavelet expansions, Bull. Amer. Math. Soc. 38 (2001), pp. 273–291; An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2003].

Fourier Transforms and Their Applications

This chapter deals with Fourier transforms in L1(ℝ) and in L2 (ℝ) and their basic properties. Special attention is given to the convolution theorem and summability kernels including Cesaro, Fejer, and Gaussian kernels. Several important results including the approximate identity theorem, general Parseval’s relation, and Plancherel theorem are proved. This is followed by the Poisson summation formula, Gibbs’ phenomenon, the Shannon sampling theorem, and Heisenberg’s uncertainty principle. Many examples of applications of the Fourier transforms to mathematical statistics, signal processing, ordinary differential equations, partial differential equations, and integral equations are discussed. Included are some examples of applications of multiple Fourier transforms to important partial differential equations and Green’s functions.

Wave packet frames on local fields of positive characteristic

The objective of this paper is to construct wave packet frames on local fields of positive characteristic. A necessary and sufficient condition for the wave packet system D p j T u ( n ) a E u ( m ) b ? j ? Z , m , n ? N 0 to be a frame for L 2 ( K ) is given by means of the Fourier transform.

TIGHT WAVELET FRAMES GENERATED BY THE WALSH POLYNOMIALS

Tight wavelet frames and their promising features in applications have attracted a great deal of interest and effort in recent years. In this paper, we give an explicit construction of tight wavelet frames generated by the Walsh polynomials on positive half-line ℝ+ using the extension principles. Finally, we derive the wavelet frame decomposition and reconstruction formulas which are similar to those of orthonormal wavelets on positive half-line ℝ+.

Explicit construction of M-band tight framelet packets

Abstract The advantages of framelet packets and their promising features in various applications have attracted a lot of interest and effort in recent years. In this paper, we present an explicit construction of M-band tight framelet packets via unitary extension principle.

Periodic Wavelet Frames on Local Fields of Positive Characteristic

ABSTRACT Periodic wavelet frames have gained considerable popularity in recent years, primarily due to their substantiated applications in diverse and widespread fields of science and engineering. In this article, we introduce the concept of periodic wavelet frame on local fields of positive characteristic and show that under weaker conditions the periodization of any wavelet frame constructed by the unitary extension principle with dilation factor 𝔭−1 is a periodic wavelet frame on local fields. Moreover, based on the mixed extension principle and Fourier-based techniques of the wavelet frames, we present an explicit method for a pair of dual periodic wavelet frames on local fields of positive characteristic.

Semi-orthogonal wavelet frames on local fields

Abstract We investigate semi-orthogonal wavelet frames on local fields of positive characteristic and provide a characterization of frame wavelets by means of some basic equations in the frequency domain. The theory of frame multiresolution analysis recently proposed by Shah [20] on local fields is used to establish equivalent conditions for a finite number of functions ψ 1 , ψ 2 , … , ψ L