Stable approximation of fractional derivatives of rough functions

Abstract

The process of numerical fractional differentiation is well known to be an ill-posed problem, and it has been discussed by many authors, and a large number of different solution methods has been proposed. However, available approaches require a knowledge of a bound of the second or third derivatives of the function under consideration which are not always available. In this paper the following mollification method is suggested for this problem: if the data are given inexactly then we mollify them by elements of well-posedness classes of the problem, namely, by elements of an appropriater-regular multiresolution approximation {V j }j ∈ ℤ ofL2(ℝ). WithinV j the problem of fractional differentiation is well-posed, and we can find a mollification parameterJ depending on the noise level ɛ in the data, such that the error estimation is of Holder type. The method can be numerically implemented by fundamental results by Beylkin, Coifman and Rokhlin ([2]) on representing differential operators in wavelet bases. It is worthwhile to note that there are very few papers concerning the question of stable numerical fractional differentiation of very rough functions. It is interesting that by our method, in a certain sense, we can approximate the derivatives of very rough functions (functions fromH s (ℝ),s e ℝ) which have no derivative in the classical sense, like the hat functions, step functions..., in a stable way. Our method is of interest, since it is local. This means that to approximate the fractional derivative of a function at a point with improperly given data, we need only local information about this function in an appropriate neighbourhood of this point.