  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.
