In this paper, we establish an one-to-one mapping between complex-valued functions defined on $R^{+}\cup \left\{0\right\}$ and complex-valued functions defined on $p$-adic number field $Q_p$, and introduce the definition and method of Weyl-Heisenberg frame on hormonic analysis to $p$-adic anylysis.

Let $k$ be a finite field. Wintenberger used the field of norms to give an equivalence between a category whose objects are totally ramified abelian $p$-adic Lie extensions $E/F$, where $F$ is a local field with residue field $k$, and a category whose objects are pairs $(K,A)$, where $K\cong k\left(\right(T\left)\right)$ and $A$ is an abelian $p$-adic Lie subgroup of ${\mathrm{Aut}}_k\left(K\right)$. In this paper we extend this equivalence to allow $\mathrm{Gal}(E/F)$ and $A$ to be arbitrary abelian pro-$p$ groups.

We prove an inequality linking the growth of a generalized Wronskian of m p-adic power series to the growth of the ordinary Wronskian of these m power series. A consequence is that if the Wronskian of m entire p-adic functions is a non-zero polynomial, then all these functions are polynomials. As an application, we prove that if a linear differential equation with coefficients in ℂₚ[x] has a complete system of solutions meromorphic in all ℂₚ, then all the solutions of the differential equation are...