A derivative on the field of \(p\)-adic numbers
The authors introduce a new pointwise \(p\)-adic derivative for complex-valued locally constant functions defined on the field \(\mathbb{Q}_p\) of \(p\)-adic numbers, and investigate some basic properties of this derivative (\(p\) is a prime number). They study the derivative of a regular distribution and of the multiplicative convolution, as well as the derivative of the Fourier transform. Among other things, they prove that the Fourier transform of any locally constant function with compact support is infinitely differentiable. The results obtained in this paper provide a partial answer to the problem posed in [\textit{W.-Y. Su} and \textit{H. Qiu}, ``\(p\)-adic calculus and its applications to fractal analysis and medical science'', Facta Univ., Ser. Electron. Energ. 21, No. 3, 339--347 (2008; \url{doi:10.2298/FUEE0803339S})] of developing \(p\)-adic calculus on the multiplicative group of a \(p\)-adic field.
scientific article; zbMATH DE number 6105052
scientific article