Abstract
We use a previously obtained topological characterization of Gelfand-Shilov spaces of Beurling type to characterize its dual using Riesz representation theorem. Using the characterization of the dual space equipped with the weak topology, we study the action of Ornstein-Uhlenbeck Semigroup on the dual space.
Highlights
In this paper, we use the characterization of Gelfand-Shilov spaces of Beurling type of test functions of tempered ultradistribution in terms of their Fourier transform obtained in [1] to prove structure theorem for functionals in dual space (Σβα)′
The Fourier transform of a function f will be denoted by F (f ) or f and it will be defined as Rn e−2πixξf (x) dx
Chung et al proved symmetric characterizations for Gelfand-Shilov spaces via the Fourier transform in terms of the growth of the function and its Fourier transform which imposes no conditions on the derivative
Summary
E−N|x|1/α dx < ∞, for all α > 1, Rn where b is the constant in Property 2 of Remark 2.2. Property 1 in Remark 2.2 implies that ||1/α is subadditive. ([5])Given a functional L in the topological dual of the space C0, there exists a unique regular complex Borel measure μ so that. Given L ∈ Σβα → C, the following statements are equivalent : (i) L ∈ (Σβα)′ (ii) There exist two regular complex Borel measures μ1 and μ2 of finite total variation and k ∈ N0 such that. Proof: Proving (i) implies (ii): Given L ∈ (Σβα)′, there exist constants k and C so that. 3.1, there exist regular complex Borel measures μ1 and μ2 of finite total variation so that. Proving (ii) implies (i): If μ1 and μ2 are regular complex Borel measures satisfying (ii) and φ ∈ Σβα, ek|ξ|1/β φdμ
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