Some problems in the theory of approximation of functions on the group of \(p\)-adic numbers
On a locally compact group, such as the complex numbers \(\mathbb{C}\), the Haar measure (normalized to take value \(1\) on \(\mathbb{Z}_p\)) is at one's disposal. In particular, it permits integrating functions \(f : \mathbb{Q}_p \to \mathbb{C}\) on the \(p\)-adic numbers. This paper studies the function spaces \[ L^\rho(\mathbb{Q}_p) := \left\{ f : \mathbb{Q}_p \to \mathbb{C} : \int_{\mathbb{Q}_p} |f|^\rho < \infty\right \} \] for \(1 \leq \rho \leq \infty\). For example, -- for \(\rho = 1\), these are the functions such that \[ \int_{\mathbb{Q}_p} |f| < \infty, \] -- for \(\rho = \infty\), these are the uniformly bounded functions. When a function is not locally constant, the notion of a Lipschitz-function captures the intuition that it varies little locally: While for \(\rho = \infty\) the definition of a Lipschitz-function in \(L^\rho(\mathbb{Q}_p)\) of order \(\alpha\) is the usual one of uniform boundedness \(|f(x) - f(y)| \leq C \cdot |x-y|^\alpha\) for a constant \(C > 0\), it is generalized to \(0 < \rho < \infty\) by uniformly bounding the integrals \[ \| f - \tau_h f \|_\rho=\int |f - \tau_h f|^\rho \] for translation operators \(\tau_h : f \mapsto f(\cdot + h)\) with \(h \to 0\) \(p\)-adically: 1. More exactly, the modulus of continuity \[ \omega(n, \rho) := \sup \{\| f - \tau_h f \|_\rho : |h| \leq p^{-n} \} \] measures how far \(f\) is from local constancy. If there is a constant \(C > 0\) such that \(\omega(n, \rho) \leq C \cdot p^{-n \alpha}\) for all \(n\), then \(f\) is a Lipschitz-function of order \(\alpha\). Theorem 1.5 shows that a function in \(L^\rho(\mathbb{Q}_p)\) is Lipschitz if and only if can be uniformly approximated by locally constant functions with respect to \(\| \cdot \|_\rho\) and relates the order to the speed of convergence. 2. Theorems 1.7, respectively 1.8, give a condition on the parameters \(\alpha\), \(\rho\) and \(\beta\), \(\sigma\) for which the space of Lipschitz functions of order \(\alpha\) in \(L^\rho(\mathbb{Q}_p)\) embeds into \(L^\sigma(\mathbb{Q}_p)\) or into the space of Lipschitz functions of order \(\beta\) in \(L^\sigma(\mathbb{Q}_p)\), respectively.
scientific article; zbMATH DE number 6974393
scientific article