On uniform differentiability and \(q\)-Mahler expansions (Q2778289)

It was proved by \textit{K. Conrad} [Adv. Math. 153, 185-230 (2000; Zbl 1003.11055)] that the \(q\)-binomial coefficient functions \(\binom xn_q\), where \(q\in \mathbb Z_p\), \(|q-1|_p<1\), form an orthonormal basis in the space of continuous functions on \(\mathbb Z_p\) with values from an extension \(K\) of \(\mathbb Q_p\). A characterization of the expansion coefficients for differentiable functions was also given. NEWLINENEWLINENEWLINEThe authors introduce the notion of a strictly differentiable function \(f:\;\mathbb Z_p\to K\). That means that \((x-y)^{-1}[f(x)-f(y)]\) has a limit as \((x,y)\to (a,a)\), for every \(a\in \mathbb Z_p\). Let \(C^{(m)}(\mathbb Z_p,K)\) be the space of \(m\) times strictly differentiable functions. Their characterization is given in terms of coefficients of \(\binom xn_q\)-expansions. It is shown that the \(q\)-binomial coefficient functions form an orthonormal basis in \(C^{(m)}(\mathbb Z_p,K)\) with respect to the natural norm.