Norms of Schneider--Teitelbaum Polynomials in $p$-adic Fourier Theory

Let $F$ be the unramified quadratic extension of $\mathbb{Q}_p$, and let $P_n(T)$ denote the polynomials arising in the $p$-adic Fourier theory of Schneider and Teitelbaum. We determine explicitly the norms of the functions $P_n(xΩ)$ on the Banach spaces of locally $F$-analytic functions of fixed order, where $Ω$ is a Lubin--Tate period. Our computation is based on a recent valuation formula of Ardakov and Berger for the values $P_n(Ω)$. As an application, we obtain an explicit normalization of the basis constructed by Bannai and Kobayashi, whose elements all have norm one.