Q-binomials and non-continuity of the p-adic Fourier transform

TL;DR

This paper investigates the behavior of Schwartz functions over finite extensions of p-adic fields, demonstrating that any such function can be approximated by functions whose Fourier transforms vanish uniformly, using q-binomial coefficients.

Contribution

It introduces a novel approach employing q-binomial coefficients to analyze the non-continuity of the p-adic Fourier transform on Schwartz functions.

Findings

Any Schwartz function can be approximated by functions with Fourier transforms tending to zero.

The proof leverages classical identities involving q-binomial coefficients.

The work reveals non-continuity properties of the p-adic Fourier transform.

Abstract

Let F be a finite extension of Q_p. We show that every Schwartz function on F, with values in an algebraic closure of Q_p, is the uniform limit of a sequence of Schwartz functions, whose Fourier transforms tend uniformly to 0. The proof uses the notion of q-binomial coefficients and some classical identities between them.