Reflection Symmetry in Fractional Calculus – Properties and Applications

Abstract

In this paper we define Riesz type derivatives symmetric and anti-symmetric w.r.t. the reflection mapping in finite interval [a,b]. Functions determined in [a,b] are split into parts with well determined reflection symmetry properties in a hierarchy of intervals [a m ,b m ], m ∈ ℕ, concentrated around an arbitrary point. For these parts - called the [J]-projections of function, we prove the representation and integration formulas for the introduced fractional symmetric and anti-symmetric integrals and derivatives. It appears that they can be reduced to operators determined in arbitrarily short subintervals [a m ,b m ]. The future application in the reflection symmetric fractional variational calculus and the generalization of previous results on localization of Euler-Lagrange equations are discussed.