Holomorphic Continuation via Laplace-Fourier series

TL;DR

This paper demonstrates that under certain conditions, the Laplace-Fourier series of a smooth function on a Euclidean ball can be extended holomorphically to a Lie ball in complex space, ensuring compact convergence.

Contribution

It establishes conditions for the holomorphic extension of Laplace-Fourier series of smooth functions into complex space, expanding the understanding of their analytic continuation.

Findings

Laplace-Fourier series can be extended holomorphically to the Lie ball.

The extension converges compactly under natural coefficient estimates.

Provides a new method for analytic continuation of smooth functions.

Abstract

Let $B_{R}$ be the ball in the euclidean space $\mathbb{R}^{n}$ with center 0 and radius $R$ and let $f$ be a complex-valued, infinitely differentiable function on $B_{R}.$ We show that the Laplace-Fourier series of $f$ has a holomorphic extension which converges compactly in the Lie ball $\hat {B_{R}}$ in the complex space $\mathbb{C}^{n}$ when one assumes a natural estimate for the Laplace-Fourier coefficients.