Applications of Tauberian theorems in some problems in mathematical physics

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

doi:10.1007/s11232-008-0140-6

Cited by 11 publications

(7 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We point out that this problem was first raised and studied by Drozhzhinov and Zavialov [9,10]. Their results are robust and they demonstrated their usefulness with several very interesting applications; for example, they discovered a general abstract class of onedimensional Besov type spaces in [9], and established in [12] useful norm estimates in various Banach spaces for solutions to the Schrödinger equation. However, their theory excludes a wide number of kernels that are of great interest in harmonic analysis.…”

Section: Introductionmentioning

confidence: 86%

“…When Γ = R n and P (∂/∂x) = ∆, we obtain that stabilization along parabolas (i.e., d = 2) is sufficient for stabilization in time of the solution to the Cauchy problem for the heat equation. This particular case of Corollary 8.12 was studied in [9,10,12]. 8.3.…”

Section: Asymptotic Stabilization In Time For Cauchy Problemsmentioning

confidence: 99%

“…In this article we investigate several Tauberian aspects of integral transforms arising from regularizations of distributions. The seminal work of Drozhzhinov and Zavialov [9,10,12] has been used as the starting point for our analysis. We shall connect our work with those of Meyer, Jaffard, Estrada, Holschneider and collaborators.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multidimensional Tauberian theorems for wavelet and non-wavelet transforms

Pilipović,

Vindas

2010

Preprint

View full text Add to dashboard Cite

We study several Tauberian properties of regularizing transforms of tempered distributions with values in Banach spaces, that is, transforms of the form, where the kernel ϕ is a test function and ϕ y (•) = y −n ϕ(•/y). If the zeroth moment of ϕ vanishes, it is a wavelet type transform; otherwise, we say it is a non-wavelet type transform.The first aim of this work is to show that the scaling (weak) asymptotic properties of distributions are completely determined by boundary asymptotics of the regularizing transform plus natural Tauberian hypotheses. Our second goal is to characterize the spaces of Banach space-valued tempered distributions in terms of the transform M f ϕ (x, y). We investigate conditions which ensure that a distribution that a priori takes values in locally convex space actually takes values in a narrower Banach space. Special attention is paid to find the optimal class of kernels ϕ for which these Tauberian results hold.We give various applications of our Tauberian theory in the pointwise and (micro-)local regularity analysis of Banach spacevalued distributions, and develop a number of techniques which are specially useful when applied to scalar-valued functions and distributions. Among such applications, we obtain the full weakasymptotic series expansion of the family of Riemann-type distributions R β (x) = ∞ n=1 e iπxn 2 /n 2β , β ∈ C, at every rational point. We also apply the results to regularity theory within generalized function algebras, to the stabilization of solutions for a class of Cauchy problems, and to Tauberian theorems for the Laplace transform; in addition, we find a necessary and sufficient condition for the existence of f (t 0 , ξ) ∈ S ′ (R n ξ ), where f (t, ξ) ∈ S ′ (R n t × R n ξ ).

show abstract

Section: Introductionmentioning

confidence: 86%

Section: Asymptotic Stabilization In Time For Cauchy Problemsmentioning

confidence: 99%

See 1 more Smart Citation

Multidimensional Tauberian theorems for wavelet and non-wavelet transforms

Pilipović,

Vindas

2010

Preprint

View full text Add to dashboard Cite

show abstract

“…Multidimensional Tauberian theorems were then systematically investigated by him, Drozhzhinov, and Zav'yalov, and their approach resulted in a powerful Tauberian machinery for multidimensional Laplace transforms of Schwartz distributions. Such results have been very useful in probability theory [30] and mathematical physics [1,11,29]. Tauberian theorems for other integral transforms of generalized functions have been extensively studied by several authors as well, see e.g.…”

Section: Introductionmentioning

confidence: 99%

A multidimensional Tauberian theorem for Laplace transforms of ultradistributions

Neyt

Vindas

2019

Integral Transforms and Special Functions

View full text Add to dashboard Cite

show abstract

“…This Tauberian question was raised and studied by Drozhzhinov and Zav'yalov in one dimension in [8] and in several variables in [9], where they demonstrated that many Tauberian theorems for integral transforms become particular instances of this problem. See also [10] for several other interesting applications. In this article we revisit the problem and provide optimal results by finding the largest class of test functions for which the space of E-valued distributions (up to some correction terms) admits a characterization in terms of a class estimate.…”

Section: Introductionmentioning

confidence: 99%

Tauberian class estimates for vector-valued distributions

Pilipović¹,

Vindas²

2019

Sb. Math.

View full text Add to dashboard Cite

We study Tauberian properties of regularizing transforms of vectorvalued tempered distributions, that is, transforms of the form M f ϕ (x, y) = (f * ϕ y )(x), where the kernel ϕ is a test function and ϕ y (·) = y −n ϕ(·/y). We investigate conditions which ensure that a distribution that a priori takes values in locally convex space actually takes values in a narrower Banach space. Our goal is to characterize spaces of Banach space valued tempered distributions in terms of so-called class estimates for the transform M f ϕ (x, y). Our results generalize and improve earlier Tauberian theorems of Drozhzhinov and Zav'yalov (Sb. Math. 194 (2003), 1599-1646. Special attention is paid to find the optimal class of kernels ϕ for which these Tauberian results hold.Dedicated to the memory of Vasiliȋ Sergeevich Vladimirov and Boris Ivanovich Zav'yalov 2010 Mathematics Subject Classification. 40E05, 46F05, 46F12.

show abstract

Applications of Tauberian theorems in some problems in mathematical physics

Cited by 11 publications

References 11 publications

Multidimensional Tauberian theorems for wavelet and non-wavelet transforms

Multidimensional Tauberian theorems for wavelet and non-wavelet transforms

A multidimensional Tauberian theorem for Laplace transforms of ultradistributions

Tauberian class estimates for vector-valued distributions

Contact Info

Product

Resources

About