Some peculiarities of transition from discrete to continuum Fourier series in lattice theories

TL;DR

This paper investigates the transition from discrete to continuous Fourier series in lattice theories, highlighting conditions where replacing sums with integrals is invalid, especially for singular functions.

Contribution

It identifies specific conditions under which the summation-to-integration replacement fails in the transition from discrete to continuum Fourier series.

Findings

Conditions for invalid sum-to-integral replacement

Analysis of singular functions in Fourier series transition

Guidelines for proper continuum limit handling

Abstract

Transition from discrete to continuous Fourier series is studied for the functions becoming singular in the transition. Conditions are specified when summing replacement by integration is inadmissible.