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Asymptotic Fourier and Laplace transformations for hyperfunctions. (English) Zbl 1225.44001

Author’s abstract: We develop an elementary theory of Fourier and Laplace transformations for exponentially decreasing hyperfunctions. Since any hyperfunction can be extended to an exponentially decreasing hyperfunction, this provides simple notions of asymptotic Fourier and Laplace transformations for hyperfunctions, improving the existing models.
This is used to prove criteria for the uniqueness and solvability of the abstract Cauchy problem in Fréchet spaces.

MSC:

44A10 Laplace transform
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
46F15 Hyperfunctions, analytic functionals
34G10 Linear differential equations in abstract spaces
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