Tauberian theorem for generalized multiplicative convolutions

Drozhzhinov, Yu. N.; Zav'yalov, B. I.

doi:10.1070/im2000v064n01abeh000274

Cited by 8 publications

(6 citation statements)

References 10 publications

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“…The Tauberian condition which we shall use may be referred as a Vladimirov-DrozhzhinovZavialov type Tauberian condition. Actually, they have made extensive use of these types of conditions in the study of Tauberian theorems for local behavior of generalized functions in terms of several integral transforms, see [7,8,47,48].…”

Section: Definitionmentioning

confidence: 99%

“…The Lojasiewicz notion admits a natural generalization, the quasiasymptotic behavior, which may be used to describe pointwise asymptotic properties of distributions as well as asymptotic properties at infinity. The quasiasymptotics were introduced by Zavialov [54] as a result of his investigations in quantum field theory, and further developed by him, Vladimirov and Drozhzhinov (see [7]- [10], [47]- [50]). Later on, the theory had its main developments within the study of integral transforms, convolution equations, partial differential equations, multiresolution expansions and Abelian and Tauberian theory (see [7]- [10], [13]- [15], [30]- [33], [38], [47]- [53]).…”

Section: Introductionmentioning

confidence: 99%

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Tauberian Theorems for the Wavelet Transform

Vindas

Pilipović

Rakić

2010

J Fourier Anal Appl

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Abstract. We make a complete wavelet analysis of asymptotic properties of distributions. The study is carried out via Abelian and Tauberian type results, connecting the boundary asymptotic behavior of the wavelet transform with local and non-local quasiasymptotic properties of elements in the Schwartz class of tempered distributions. Our Tauberian theorems are full characterizations of such asymptotic properties. We also provide precise wavelet characterizations of the asymptotic behavior of elements in the dual of the space of highly time-frequency localized functions over the real line. For the use of the wavelet transform in local analysis, we study the problem of extensions of distributions initially defined on R \ {0} to R; in this extension problem, we explore the asymptotic properties of extensions of a distribution having a prescribed asymptotic behavior. Our results imply intrinsic properties of functions and measures as well, for example, we give a new proof of the classical Littlewood Tauberian theorem for power series.

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Section: Definitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Tauberian Theorems for the Wavelet Transform

Vindas

Pilipović

Rakić

2010

J Fourier Anal Appl

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“…We will prove (13) similarly as (6). It is sufficient to prove that, for every σ ∈ Λ and ε > 0, lim h→∞ H σγ (h) = 0 (γ ∈ P * ) where…”

Section: N)mentioning

confidence: 96%

“…[7,10,15]). Essential extensions in this sense were obtained in [6,18] for distributions and in [19] for ultradistributions.…”

Section: Introductionmentioning

confidence: 98%

Wiener-Type Tauberian Theorems for Fourier Hyperfunctions

Pilipović¹,

Stanković²

2002

Z. Anal. Anwend.

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“…In Wiener Tauberian theory [55] and its many extensions [1,4,27,28] the Tauberian kernels are those whose Fourier transforms do not vanish at any point. In our theory the Tauberian kernels will be those ϕ such thatφ does not identically vanish on any ray through the origin.…”

Section: Introductionmentioning

confidence: 99%

Multidimensional Tauberian theorems for vector-valued distributions

Pilipović

Vindas

2014

Publ Inst Math (Belgr)

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We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of $f$ is given by the integral transform $M^{f}_{\varphi}(x,y)=(f\ast\varphi_{y})(x),$ $(x,y)\in\mathbb{R}^{n}\times\mathbb{R}_{+}$, with kernel $\varphi_{y}(t)=y^{-n}\varphi(t/y)$. We apply our results to the analysis of asymptotic stability for a class of Cauchy problems, Tauberian theorems for the Laplace transform, the comparison of quasiasymptotics in distribution spaces, and we give a necessary and sufficient condition for the existence of the trace of a distribution on $\left\{x_0\right\}\times \mathbb R^m$. In addition, we present a new proof of Littlewood's Tauberian theorem.Comment: 28 pages. arXiv admin note: substantial text overlap with arXiv:1012.509

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Tauberian theorem for generalized multiplicative convolutions

Cited by 8 publications

References 10 publications

Tauberian Theorems for the Wavelet Transform

Tauberian Theorems for the Wavelet Transform

Wiener-Type Tauberian Theorems for Fourier Hyperfunctions

Multidimensional Tauberian theorems for vector-valued distributions

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