Stability results for fractional differentiation

Abstract

Let α = (α₁,α₂,…,α_n) be a multi index with α_i>0,i = 1,2,…,n,and let I^α be the Liouville fractional integral operator in IRⁿ: I^αu(x)=

with Г(z) being the Euler function. A method for solving the ill posed problem

in a stable way is proposed. As a consequence, a stability estimate of Hölder type is established. The method gives regularized solutions which are entire functions of exponential type and converge to the exact solution not only in L_q norm but also pointwise where it is continous. Similar related results for Riesz and Bessel fractional integral operators are also outlined

Keywords:

• Fractional differentiation
• ill-posed
• regularization
• mollification

AMS:

• 45D05
• 65R30

^∗Present address: FB Mathematik, Johann Wolfgang Goethe-Universitat Frankfurt am Main, D-60054 Frankfurt, Germany.

^∗Present address: FB Mathematik, Johann Wolfgang Goethe-Universitat Frankfurt am Main, D-60054 Frankfurt, Germany.

Notes

^∗Present address: FB Mathematik, Johann Wolfgang Goethe-Universitat Frankfurt am Main, D-60054 Frankfurt, Germany.