The continuous wavelet transform and window functions

Abstract

We define a window function ψ as an element of L 2 (R n ) satisfying certain boundedness properties with respect to the L 2 (R n ) norm and prove that it satisfies the admissibility condition if and only if the integral ofalong the real line is zero. We also prove that each of the window functions is an element of L 1 (R n ). A function ψ ∈ L 2 (R n ) satisfying the admissibility condition is a wavelet. We define the wavelet transform of f ∈ L 2 (R n ) (which is a window function) with respect to the wavelet ψ ∈ L 2 (R n ) and prove an inversion formula interpreting convergence in L 2 (R n ). It is also proved that at a point of continuity of f the convergence of our wavelet inversion formula is in a pointwise sense.

A certain family of fractional wavelet transformations

“…Proof By applying the technique, which was used earlier Pandey and Upadhyay, we can show that which, in light of , yields We thus find that …”

Continuous Wavelet Transform of Schwartz Tempered Distributions in S′ ( R n )

“…As studied in the earlier works (see, for example, [1][2][3][4][5][6][7][8][9][10][11][12], we define a Schwartz testing function space S(R n ) to consist of C ∞ functions φ defined on R n and satisfying the following conditions:…”

“…is a wavelet belonging to S(R n ). Let s(R n ) be a subspace of S(R n ) such that every element φ ∈ s(R n ) satisfies Equation (4). Clearly, every element of s(R n ) is a wavelet [4].…”

“…The work on the multidimensional wavelet transform with positive scale [a > 0] was done by Daubechies [15], Meyer [16], Pathak [17], and some others. Motivated by the earlier works [6,8,12], Pandey et al [4] studied a generalization of these works and extended the multidimensional wavelet transform with real scale [a = 0]. In the year 1995, Holschneider [18] extended the multidimensional wavelet transform to Schwartz tempered distributions with positive scales [a > 0].…”

“…The main advantage of our work is a possible further practical utility of the proven result and the simplicity of calculation; in addition, our extension of the multidimensional wavelet inversion formula is the most general. In [4], it is proved that a window function ψ(x) ∈ L 2 (R n ) is a wavelet if and only if the integral of ψ along each of the axes is zero; therefore, any ψ(x) ∈ s(R n ) is a wavelet. Hence, the wavelet transform of a constant distribution is zero.…”