# A remark on convergence almost-everywhere of eigenfunction expansions of elliptic operators

## TL;DR

This paper introduces a simple method to improve convergence theorems for eigenfunction expansions of elliptic operators, including new results on spherical Fourier series convergence for smooth functions.

## Contribution

It proposes a novel, straightforward approach to estimate the maximal operator in L1, enhancing existing convergence theorems for eigenfunction expansions of elliptic operators.

## Key findings

- Improved convergence theorems for eigenfunction expansions
- New results on spherical partial sums of Fourier series
- Enhanced understanding of convergence almost-everywhere

## Abstract

In this paper it is proposed a very simple method for estimating the maximal operator in $L_1$. Using this method one can considerably improve the existing theorems on convergence almost-everywhere of eigenfunction expansions of an arbitrary elliptic differential operators with a point spectrum. In particular, it is obtained a new result on convergence almost-everywhere of spherical partial sums of the multiple Fourier series of smooth functions.

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Source: https://tomesphere.com/paper/1902.02035