Mollification of fractional derivatives using rapidly decaying harmonic wavelet

Abstract

We study the use of the wavelet expansion in terms of Meyer's wavelet to the problem of mollification in the numerical calculation of fractional derivative. It is shown that, when the simplest of Meyer's wavelet is used, the expansion is equivalent to the transform by the de la Vallée Poussin kernel, that was proposed by Hào et al. It is expected that better results are attained by using the rapidly decaying harmonic wavelet, which is another of Meyer's wavelet. We examine this. It is also shown that there exists an estimation of the error of approximation in the mollification by using only simple inequalities, without sophisticated inequalities which are invoked in the paper by Hào et al. Examples of numerical calculation are given.MSC 2010 : 26A33, 65D25, 65J20, 42B35

Mollification Based on Wavelets

“…It was shown that the use of the simplest of Meyer's wavelets agrees with the use of de la Vallée Poussin kernel adopted by Hào et al [2]. In [2,3], the calculation of fractional derivative D λ f (x) for λ > 0 was studied, which is generally an ill-posed problem. When f ϵ involving noise is given in place of smooth f , we calculate D λ [M f ϵ ](x), which approximates D λ f (x).…”

“…When f ϵ involving noise is given in place of smooth f , we calculate D λ [M f ϵ ](x), which approximates D λ f (x). In [2,3], we estimate how the error of this approximation can be reduced by the choice of the scale on which µ depends.…”

“…Here ν is the scale at the stage of truncation. In [3], it was shown that an average of the expansion is expressed as Equation (1) if we put µ(x) = µ ν (x) where…”

“…In a preceding paper [3], the present authors proposed to use the expansion in the orthogonal set of the rapidly decaying harmonic wavelets (rdH-wavelets), which were developed in [4,5]. The wavelets are characterized by a scaling function that we denote by ϕ 1 .…”

“…In [3], a series of Meyer's wavelets ( [6] [p. 49]), which are special ones of the rdH-wavelets, are studied. It was shown that the use of the simplest of Meyer's wavelets agrees with the use of de la Vallée Poussin kernel adopted by Hào et al [2].…”