Unbounded Transforms and Approximation of Functions over $p$-adic Fields

We consider functions of a $p$-adic variable with values in different spaces. In each case we consider an unbounded integral operator and a corresponding issue. More precisely, we study the Riesz–Volkenborn integral representation of functions with values in a non-Archimedean field, the Vladimirov operator and corresponding vectors of exponential type in spaces of complex-valued functions, and the Fourier transform and its (dis)continuity in spaces of Banach-valued functions.