Splines and fractional differential operators

Adjoint Fractional Differential Expressions and Operators

In this article we present the notions of adjoint differential expressions for fractional-order differential expressions, adjoint boundary conditions for fractional differential equations, and adjoint fractional-order operators. These notions are based on new formulas obtained for various types of fractional derivatives. The introduced notions can be used in many fields of modelling and control...

Splines: on scale, differential operators and fast algorithms

Fractional Splines and Wavelets

We extend Schoenberg’s family of polynomial splines with uniform knots to all fractional degrees α > −1. These splines, which involve linear combinations of the one-sided power functions x+ = max(0, x) α, are α-Hölder continuous for α > 0. We construct the corresponding B-splines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains....

Fractional Splines , Wavelet Bases

The purpose of this presentation is to describe a recent family of basis functions—the fractional B-splines—which appear to be intimately connected to fractional calculus. Among other properties, we show that they are the convolution kernels that link the discrete (finite differences) and continuous (derivatives) fractional differentiation operators. We also provide simple closed forms for the ...

Fractional embedding of differential operators and Lagrangian systems

— This paper is a contribution to the general program of embedding theories of dynamical systems. Following our previous work on the Stochastic embedding theory developed with S. Darses, we define the fractional embedding of differential operators and ordinary differential equations. We construct an operator combining in a symmetric way the left and right (Riemann-Liouville) fractional derivati...