Wavelet transform of Fractal Interpolation Function

TL;DR

This paper investigates the wavelet transform of Fractal Interpolation Functions (FIF), establishing their regularity and Lipschitz class membership through methods involving functional equations and Fourier transforms.

Contribution

It introduces two methods for analyzing the wavelet transform of FIF and determines their regularity and Lipschitz class under specific parameter conditions.

Findings

FIF belongs to Lipschitz class of order δ under certain conditions.

Fourier transform of FIF is derived to analyze regularity.

Wavelet transform of FIF is obtained via two different approaches.

Abstract

In the present paper, the wavelet transform of Fractal Interpolation Function (FIF) is studied. The wavelet transform of FIF is obtained through two different methods. The first method uses the functional equation through which FIF is constructed. By this method, it is shown that the FIF belongs to Lipschitz class of order $\delta$ under certain conditions on free parameters. The second method is via Fourier transform of FIF. This approach gives the regularity of FIF under certain conditions on free parameters. Fourier transform of a FIF is also derived in this paper to facilitate the approach of wavelet transform of a FIF via Fourier transform.