A Study of Wavelets on Positive Half Line

The trigonometric Fourier series has played very significant role in solving problems of science and technology. The concept of non-trigonometric Fourier series such as Haar Fourier series and Walsh-Fourier series were introduced by Haar [12] and Walsh [21], respectively; Kaczmarz et al. studied some aspects of Walsh system between 1929 and 1931. Nowadays, Paley's modification which is defined as the product of Rademacher functions is known as the Walsh function [16]. A major breakthrough came in N. J. Fine's dissertation of PhD submitted to Pennsylvania University in 1946 which was published subsequently in Transactions of American Mathematical Society in 1949. Fine introduced the concept of dyadic group G and proved that Walsh functions are its characters. Walsh [21] observed that Haar and Walsh systems are Hadamard transforms of each other. The Walsh and Haar systems perform all usual applications of orthonormal systems, for example, data transmission, multiplexing, filtering, image enhancement, and pattern recognition. The Walsh system {w_n: n ∈ ℤ₊} on ℝ₊ is defined as follows:

x ∈ ℝ₊, where ℝ₊ = [0, ∞),

where the v_j are the coefficients of the decomposition

and the function w₁(x) is defined on [0, 1] by the formula

and is extended to ℝ₊ by periodicity: w₁ (x + 1) = w₁ (x) for all x ∈ ℝ₊.

Walsh polynomials are finite linear combinations of the Walsh functions. An arbitrary Walsh polynomial of order n can be written in the following form:

where c_j are complex coefficients.

Every function f(x) which is periodic with period 1 and Lebesgue integrable f(x) on [0, 1) can be expanded in a Walsh–Fourier series.

where , n = 0, 1, 2…..

Many properties of Walsh–Fourier series are analogous to trignometric Fourier series, for example, the Dirichlet kernels for the Walsh system and trigonometric system have the same order of growth.

The study of Walsh–Fourier analysis is presented in quite detail in references Schip et al. [17], Siddiqi [18], Golubov et al. [11], Maqusi [13] and Fridli et al. [19].

The theory of wavelet has been invented and developed in early 1980s notably by Geophysicist Morlet, Er Mallat, Mathematician Meyer & Coifmann and physicist Daubechies. For detail discussion of this theme, see references Daubechies [2], Christensen [1], Walnut [20], Siddiqi [19]. In recent papers Farkov and colleagues [4–6] have studied wavelets defined by Walsh functions. Recently Farkov et al. [8] have completed a monograph on this theme, see reference (Farkov, Man-chanda, and Siddiqi). The goal of this paper is to draw attention on this theme. Furthermore, the attention on the paper of Goginava [10], in which he has studied some convergence and divergence properties of subsequences of logarithmic means of Walsh–Fourier series, is focused. It is related to the work of Moricz and Siddiqi [15]. It may be observed that the results of Goginava can be extended to approximation by Logarithmic Norlund means in Dyadic Homogeneous Banach spaces and Hardy spaces Fridli et al. [8, Chapter 2].

In Section 2, we present extension of Theorem [10] for Dyadic Homogeneous Banach Spaces. The proof of this theorem will be published separately. A resume of properties of wavelets on positive half line is presented in Section 3. In Section 4, we introduced wavelets and wavelet packet associated with non-uniform multi-resolution analysis on positive half line.