Periodic hyperfunctions and Fourier series

Abstract

Every periodic hyperfunction is a bounded hyperfunction and can be represented as an infinite sum of derivatives of bounded continuous periodic functions. Also, Fourier coefficients c α c_{\alpha } of periodic hyperfunctions are of infra-exponential growth in R n \mathbb {R}^{n} , i.e., c α > C ϵ e ϵ | α | c_{\alpha }> C_{\epsilon }e^{\epsilon |\alpha |} for every ϵ > 0 \epsilon >0 and every α ∈ Z n \alpha \in \mathbb {Z}^{n} . This is a natural generalization of the polynomial growth of the Fourier coefficients of distributions. To show these we introduce the space B L p \mathcal {B}_{L^{p}} of hyperfunctions of L p L^{p} growth which generalizes the space D L p ′ \mathcal {D}’_{L^{p}} of distributions of L p L^{p} growth and represent generalized functions as the initial values of smooth solutions of the heat equation.