Every positive definite Fourier hyperfunction is a Fourier transform of a positive and infra-exponentially tempered measure, which is the generalized Boehner-Schwartz theorem for the Fourier hyperfunctions. To prove this we characterize the distributions with exponential growth via the heat kernel method.
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Soon-Yeong Chung, Dohan Kim, Distributions with Exponential Growth and Boehner-Schwartz Theorem for Fourier Hyperfunctions. Publ. Res. Inst. Math. Sci. 31 (1995), no. 5, pp. 829–845DOI 10.2977/PRIMS/1195163720