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Although the study of functional calculus has already established necessaryand sufficient conditions for operators to be fractionalized, this paper aimsto use our well-conceived notion of integer powers of operators to constructnon-integer powers of operators. In doing so, we not only provide a moreintuitive understanding of fractional theories, but also provide a frameworkfor producing new fractional theories. Such interpolations allow one toapproximate fractional powers by finite sums of integer powers of operators,and thus may find much use in numerical analysis of fractional PDEs and thetime-frequency analysis of the fractional Fourier transform. Further, theseresults provide an interpolation procedure for the Riemann zeta function.

Fractional operators via analytic interpolation of integer powers