Holomorphic Continuation via Laplace-Fourier series

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.

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