Compact Families of Locally Convex Topological Vector Spaces, Fréchet-Schwartz and Dual Fréchet-Schwartz Spaces

Zharinov, V. V.

doi:10.1070/rm1979v034n04abeh002963

Cited by 40 publications

(25 citation statements)

References 15 publications

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“…We now give some topological properties of the spaces of Φ-ultradifferentiable functions on a compact set K defined above. Here we use some notions and results of Zharinov's survey [24] without extra references.…”

Section: § 1 Statement Of the Main Resultsmentioning

confidence: 99%

Analytic implementation of the duals of some spaces of infinitely differentiable functions

Abanin

Filip’ev

2006

Sib Math J

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Using the Fourier-Laplace transform for functionals, we describe the duals of some spaces of the infinitely differentiable functions given on convex compact sets or convex domains in R N and such that the growth of their derivatives is determined by weight sequences of a general form.A weight function (or a weight) is an arbitrary nondecreasing function ϕ : [0, ∞) → R that is convex from some t 0 ≥ 0 on and enjoys the following conditions:(Denote the class of all weights by W.Let K be a compact set in R N whose nonempty interior int K has the closure coinciding with K. Given a weight ϕ, define the Banach spaceHere and in the sequel we use the following notations: C ∞ (K) is the set of all infinitely differentiable functions on a compact set K; N 0 := N ∪ {0}; f (α) is the derivative of f corresponding to a multiindex α ∈ N N 0 ; and ϕ * (s) := sup{rs − ϕ(r) : r ≥ 0} is the monotone conjugate function of ϕ. Denote by W ↑ (W ↓ ) the collection of all sequences Φ = {ϕ n } ∞ n=1 of weight functions such that, for each n ∈ N, there is C n satisfyingfor every t ≥ 0. Given Φ in W ↑ (W ↓ ) and a compact set K in R N , we naturally define the space

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Section: § 1 Statement Of the Main Resultsmentioning

confidence: 99%

Analytic implementation of the duals of some spaces of infinitely differentiable functions

Abanin

Filip’ev

2006

Sib Math J

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“…Now let D be convex. Without loss of generality, we can assume that 0 ∈ D. Define a space We refer the reader to [24] and [16] for detailed information on these notions.…”

Section: The Function Spacesmentioning

confidence: 99%

Dual of the function algebra 𝐴^{-∞}(𝐷) and representation of functions in Dirichlet series

Abanin

Khoi

2010

Proc. Amer. Math. Soc.

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“…A Fréchet-Schwartz space (briefly, (FS) space) is one that is Fréchet and Schwartz simultaneously (see [23]). It is known (see e.g.…”

Section: Introductionmentioning

confidence: 99%

Paley-Wiener-type theorem for polynomial ultradifferentiable functions

Sharyn

2015

Carpathian Math. Publ.

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Compact Families of Locally Convex Topological Vector Spaces, Fréchet-Schwartz and Dual Fréchet-Schwartz Spaces

Cited by 40 publications

References 15 publications

Analytic implementation of the duals of some spaces of infinitely differentiable functions

Analytic implementation of the duals of some spaces of infinitely differentiable functions

Dual of the function algebra 𝐴^{-∞}(𝐷) and representation of functions in Dirichlet series

Paley-Wiener-type theorem for polynomial ultradifferentiable functions

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