Fractional operators via analytic interpolation of integer powers

Abstract

Although the study of functional calculus has already established necessary and sufficient conditions for operators to be fractionalized, this paper aims to use our well-conceived notion of integer powers of operators to construct non-integer powers of operators. In doing so, we not only provide a more intuitive understanding of fractional theories, but also provide a framework for producing new fractional theories. Such interpolations allow one to approximate fractional powers by finite sums of integer powers of operators, and thus may find much use in numerical analysis of fractional PDEs and the time-frequency analysis of the fractional Fourier transform. Further, these results provide an interpolation procedure for the Riemann zeta function.