  We will use analytic function theory and Fourier analysis to establish a
characterization for some classical umbral calculus, which will focus on the
generalization of the evaluation function. Although we cannot cover all the
umbral calculus people care about, the part about Bernoulli numbers can still
answer an open question about fractional sums raised by M\"uller and Schleicher
in 2005 and synthesize some common analysis results. We will only develop the
results which are sufficient to serve the purpose of this article, but at the
same time, we will briefly mention some possible extensions.
