  In this article, we give an explicit construction of the $p$-adic Fourier
transform by Schneider and Teitelbaum, which allows for the investigation of
the integral property. As an application, we give a certain integral basis of
the space of $K$-locally analytic functions on the ring of integers
$\mathcal{O}_K$ for any finite extension $K$ of $\mathbb{Q}_p$, generalizing
the basis constructed by Amice for locally analytic functions on
$\mathbb{Z}_p$. We also use our result to prove congruences of
Bernoulli-Hurwitz numbers at non-ordinary (i.e. supersingular) primes
originally investigated by Katz and Chellali.
