\(p\)-adic Fourier theory of differentiable functions
Let \(\mathbb{Q}_p\) be the field of \(p\)-adic numbers and let \(\mathbb{C}_p\) be the completion of an algebraic closure of \(\mathbb{Q}_p\). Let \(K/\mathbb{Q}_p\) be a finite subextension of \(\mathbb{C}_p/\mathbb{Q}_p\) of degree \(d\) and let \(\mathcal{G}(X,Y)\) be a Lubin-Tate formal group law for \(K\). Let \(\log_{\mathcal{G}}(X)\in KX\) be the logarithm of \(\mathcal{G}\) and let \(\exp(X)\in\mathbb{Q}X\) be the usual exponential series. Then there are polynomials \(P_n(Y)\in K[Y]\) such that \[ \exp(Y\log_{\mathcal{G}}(Z))=\sum_{n=0}^{\infty} P_n(Y)Z^n. \] Say that \(P_n(Y)\) is the Fourier polynomial of degree \(n\). Let \(r\) be a real number such that \(r\geq d\). There is a notion of \(r\) times differentiability for functions \(f:\mathcal{O}_K\rightarrow\mathbb{C}_p\). The main theorem of this paper states that \(f\) is \(r\) times differentiable if and only if there exist \(a_n\in\mathbb{C}_p\) such that \(f(x)=\sum_{n=0}^{\infty}a_nP_n(x)\) and \(\lim_{n\rightarrow\infty}|a_n|n^{r/d}=0\). This result had been known previously in the case \(K=\mathbb{Q}_p\).
scientific article; zbMATH DE number 6915996
scientific article