Generalization of Orthogonal Kravchenko Wavelets with Convolutions of Rectangular Pulse and Atomic Functions

Abstract

Construction of orthogonal Kravchenko wavelets is similar to Meyer wavelets. It begins from Fourier transform of scaling function. It must be supported on the segment $[- 4 \pi /3; 4 \pi /3]$ and equal to one on the segment $[- 2 \pi /3; 2 \pi /3]$. Sum of its shifted squares must present partition of unity. If Fourier transform of scaling function is selected, then scaling function and wavelet may be found by known formulae. Meyer wavelets use trigonometric polynomials as Fourier transform of scaling function. For Kravchenko wavelets square root of sums of scaled and shifted atomic functions are applied.Atomic functions are compactly supported solutions of differential equations with linearly transformed argument of special form. They are infinitely smooth. Their shifts present partition of unity, which is used for wavelet construction. Note that requirements of support and constant segment length impose strict restrictions on selected atomic functions. If atomic functions $\mathrm{h}_{a}(x)$ or ch $_{a,n}(x)$ is used then among continuum of real values of parameter a only countable set of rational values defined by number of summed shifts and positive integer n (for ch $_{a,n}(x))$ is applicable.In the report another approach to construction of squared spectrum of scaling function is demonstrated. Convolution of rectangular pulse of length 3 and arbitrary partly continuous function supported on segment of length 1 would have support of length 4 and constant segment of length 2. Shifts of obtained function would present partition of unity according to properties of convolution. Then, after appropriate scaling such function may be considered as squared Fourier transform of scaling function of Meyer-like wavelet. Convolutions of atomic functions and rectangular pulses may be simply computed by means of Fourier transform. If we consider atomic functions in convolutions of rectangular pulses we obtain generalization of Kravchenko wavelets. Some of special cases of such convolutions are equivalent to sums of scaled shifts of atomic functions. This approach allows us to overcome limitations on values of atomic function parameters.