On the non-triviality of certain spaces of analytic functions. Hyperfunctions and ultrahyperfunctions of fast growth

Debrouwere, Andreas; Vindas, Jasson

doi:10.1007/s13398-017-0392-9

Cited by 17 publications

(17 citation statements)

References 37 publications

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“…We introduce the following set of conditions on

M_{p}

and

A_{p}

M_{p} and A_{p} satisfy false(M . 1 false) and false(M . 2 false), p! ≺ M_{p} A_{p}, and {scriptS}_{(A_{p})}^{(M_{p})} true(R^{d} true) is non-trivial .

A sufficient condition for the non‐triviality of

{scriptS}_{(A_{p})}^{(M_{p})} true(R^{d} true)

p!^{σ} \subset M_{p}

and

p!^{τ} \subset A_{p}

for some

σ, τ > 0

with

σ + τ > 1

[, p. 235]. Other non‐triviality conditions can be found in . Under the general conditions , we have the ensuing properties: (i)The Fourier transform is a topological isomorphism from

{scriptS}_{†}^{*} true(R^{d} true)

onto

{scriptS}_{*}^{†} true(R^{d} true)

, where we fix the constants in the Fourier transform as follows

0true scriptF (φ) (ξ) = t...

…”

Section: Preliminariesmentioning

confidence: 99%

On weighted inductive limits of spaces of ultradifferentiable functions and their duals

Debrouwere

Vindas

2018

Mathematische Nachrichten

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In the first part of this paper we discuss the completeness of two general classes of weighted inductive limits of spaces of ultradifferentiable functions. In the second part we study their duals and characterize these spaces in terms of the growth of convolution averages of their elements. This characterization gives a canonical way to define a locally convex topology on these spaces and we give necessary and sufficient conditions for them to be ultrabornological. In particular, our results apply to spaces of convolutors for Gelfand-Shilov spaces. K E Y W O R D S completeness of inductive limits, convolution, Gelfand-Shilov spaces, short-time Fourier transform, ultrabornological (PLS)-spaces M S C ( 2 0 1 0 ) 46A13, 46E10, 46F05, 46F10

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“…We introduce the following set of conditions on

M_{p}

and

A_{p}

M_{p} and A_{p} satisfy false(M . 1 false) and false(M . 2 false), p! ≺ M_{p} A_{p}, and {scriptS}_{(A_{p})}^{(M_{p})} true(R^{d} true) is non-trivial .

A sufficient condition for the non‐triviality of

{scriptS}_{(A_{p})}^{(M_{p})} true(R^{d} true)

p!^{σ} \subset M_{p}

and

p!^{τ} \subset A_{p}

for some

σ, τ > 0

with

σ + τ > 1

[, p. 235]. Other non‐triviality conditions can be found in . Under the general conditions , we have the ensuing properties: (i)The Fourier transform is a topological isomorphism from

{scriptS}_{†}^{*} true(R^{d} true)

onto

{scriptS}_{*}^{†} true(R^{d} true)

, where we fix the constants in the Fourier transform as follows

0true scriptF (φ) (ξ) = t...

…”

Section: Preliminariesmentioning

confidence: 99%

On weighted inductive limits of spaces of ultradifferentiable functions and their duals

Debrouwere

Vindas

2018

Mathematische Nachrichten

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“…for some ε > 0; see also [7,Proposition 4.3], where it is moreover shown that this rate of decay on a strip is essentially best possible. Define h : R → C by…”

Section: Optimal Decay For Functionsmentioning

confidence: 98%

An abstract approach to optimal decay of functions and operator semigroups

Debruyne

Seifert²

2019

Isr. J. Math.

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We provide a new and significantly shorter optimality proof of recent quantified Tauberian theorems, both in the setting of vectorvalued functions and of C0-semigroups, and in fact our results are also more general than those currently available in the literature. Our approach relies on a novel application of the open mapping theorem.2010 Mathematics Subject Classification. 40E05, 47D06 (44A10, 34D05).

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“…[11, p. 235]. Other non-triviality conditions can be found in [5]. If a weight function ω is pp!q-admissible, then Assumption 3.1 is fulfilled for M p and ω, whenever M p is weight sequence that satisfies pM.1q and plog pq p ă M p , as follows from [ In the rest of this section, we fix a weight sequence M p satisfying pM.1q and pM.2q 1 and a weight function ω such that Assumption 3.1 holds.…”

Section: The Spaces DL 1 ω and 9 B 1ωmentioning

confidence: 99%

On the space of ultradistributions vanishing at infinity

Debrouwere

Neyt

Vindas

2020

Banach J. Math. Anal.

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We use methods from time-frequency analysis to study the structural and linear topological properties of the space 9 B 1ω of ultradistributions vanishing at infinity (with respect to a weight function ω). Particularly, we show the first structure theorem for 9 B 1ω under weaker hypotheses than were known so far. As an application, we determine the structure of the S-asymptotic behavior of ultradistributions.

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On the non-triviality of certain spaces of analytic functions. Hyperfunctions and ultrahyperfunctions of fast growth

Cited by 17 publications

References 37 publications

On weighted inductive limits of spaces of ultradifferentiable functions and their duals

On weighted inductive limits of spaces of ultradifferentiable functions and their duals

An abstract approach to optimal decay of functions and operator semigroups

On the space of ultradistributions vanishing at infinity

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