Abstract
Using a previously obtained structure theorem of Gelfand-Shilov spaces $\Sigma _{\alpha }^{\beta }$ of Beurling type of ultradistributions, we prove that these ultradistributions can be represented as an initial values of solutions of the heat equation by describing the action of the Gauss-Weierstrass semigroup on the dual space $(\Sigma _{\alpha }^{\beta})^{\prime }.$
Highlights
In the following theorem we state a symmetric characterization of the Gelfand-Shilov spaces Σβα in terms of the Fourier transformations
We can employ the above theorem to prove the following structure theorem for functionals T ∈ (Σβα)′
Applications we study some applications on the structure theorem of Σβα tempered ultradistributions stated in Theorem 2.4 by proving some results on a semi-group acting on the Frechet space Σβα and extend it to its dual (Σβα)′
Summary
In the following theorem we state a symmetric characterization of the Gelfand-Shilov spaces Σβα in terms of the Fourier transformations. Property 2 of Remark 2.1 implies that the function e−N|x|1/α is integrable for some positive constant Property 1 in Remark 2.1 implies that ||1/α is subadditive.. The space Σβα, equipped with the family of semi-norms
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