Asymptotical behavior of one class of $p$-adic singular Fourier integrals

Khrennikov, A. Yu.; Shelkovich, V. M.

Asymptotical behavior of one class of $p$-adic singular Fourier integrals

Authors

Description

We study the asymptotical behavior of the $p$-adic singular Fourier integrals $$ J_{π_α,m;ϕ}(t) =\bigl< f_{π_α;m}(x)χ_p(xt), ϕ(x)\bigr> =F\big[f_{π_α;m}ϕ\big](t), \quad |t|_p \to \infty, \quad t\in \bQ_p, $$ where $f_{π_α;m}\in {\cD}'(\bQ_p)$ is a {\em quasi associated homogeneous} distribution (generalized function) of degree $π_α(x)=|x|_p^{α-1}π_1(x)$ and order $m$, $π_α(x)$, $π_1(x)$, and $χ_p(x)$ are a multiplicative, a normed multiplicative, and an additive characters of the field $\bQ_p$ of $p$-adic numbers, respectively, $ϕ\in {\cD}(\bQ_p)$ is a test function, $m=0,1,2...$, $α\in \bC$. If $Reα>0$ the constructed asymptotics constitute a $p$-adic version of the well known Erdélyi lemma. Theorems which give asymptotic expansions of singular Fourier integrals are the Abelian type theorems. In contrast to the real case, all constructed asymptotics have the {\it stabilization} property.