The Peano–Sard theorem for Caputo fractional derivatives and applications

Abstract

The classical Peano–Sard theorem is a very useful result in approximation theory, bounding the errors of approximations that are exact on sets of polynomials. A fractional version was developed by Diethelm for fractional derivatives of Riemann–Liouville type, which we here extend to fractional derivatives of Caputo type. We indicate some applications to quadrature and interpolation formulae. These results will be useful in the approximate solution of fractional differential equations involving Caputo-type operators, which are often said to be more natural for applications.