Abstract
In this paper the sinc-fractional derivative is extended to the Hilbert space based on Shannon wavelets. Some new fractional operators based on wavelets are defined. One of the main task is to investigate the localization and compression properties of wavelets when dealing with the non-integer order of a differential operator.
Highlights
In recent years, fractional calculus has been growing fast both in theory and applications to many different fields
According to the suitable choice of the fractional differential operator, there follows a corresponding model of analysis so that the physical model and the corresponding physical interpretation of the results it strongly depends on the chosen fractional operator
In some recent papers [8,9,10,11,12,13,14,15] the classical Lie symmetry analysis has been combined with the Riemman-Liouville fractional derivative to solve time fractional partial differential equations
Summary
Fractional calculus has been growing fast both in theory and applications to many different fields. Among the many interesting definitions of fractional operators, some Authors have recenlty proposed a fractional differential operator based on the sinc-function [20] This function is very popular in the signal analysis, because it is a localized function with slow decay. The sinc-fractional operator will be generalized in order to compute the fractional derivative of the L2(R)-functions belonging to the Hilbert space defined by the Shannon wavelet. The differential properties of the functions belonging to the Hilbert space based on Shannon wavelet are given, together with the explicit form of the integer order derivatives (see [24, 26]).
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