Hardy type asymptotics for cosine series in several variables with decreasing power-like coefficients

TL;DR

This paper derives explicit asymptotic formulas for multi-variable cosine series with decreasing power-like coefficients near the origin, extending classical Hardy results from one dimension to higher dimensions using elementary algebraic methods.

Contribution

It introduces a simple algebraic approach to find asymptotics of multi-variable cosine series with power-like decay, filling a gap in higher-dimensional analysis.

Findings

Explicit asymptotic expressions obtained for the series

Extension of Hardy's one-dimensional results to multiple variables

Elementary method avoids complex asymptotic machinery

Abstract

The investigation of the asymptotic behavior of trigonometric series near the origin is a prominent topic in mathematical analysis. For trigonometric series in one variable, this problem was exhaustively studied by various authors in a series of publications dating back to the work of G. H. Hardy, 1928. Trigonometric series in several variables have got less attention. The aim of the work is to partially fill this gap by finding the asymptotics of trigonometric series in several variables with the terms, having a form of `one minus the cosine' up to a decreasing power-like factor: \[ \sum_{z\in\mathbb{Z}^{d}\setminus\{0\}}\frac{1}{\|z\|^{d+\alpha}}\left(1-\cos\langle z,\theta\rangle\right), \qquad \theta\in\mathbb{R}^{d}, \] where $\langle\cdot,\cdot\rangle$ is the standard inner product and $\|\cdot\|$ is the max-norm on $\mathbb{R}^{d}$. The approach developed in the paper is quite elementary and essentially algebraic. It does not rely on the classic machinery of the asymptotic analysis such as slowly varying functions, Tauberian theorems or the Abel transform. However, in our case, it allows to obtain explicit expressions for the asymptotics and to extend to the general case $d\ge 1$ classical results of G. H. Hardy and other authors known for $d=1$.