Wiener Tauberian Theorems for Distributions

2014

In this doctoral dissertation several integral transforms are discussed.The first one is the Short time Fourier transform (STFT). We present continuity theorems for the STFT and its adjoint on the test function space K1(ℝn) and the topological tensor product K1(ℝn) ⊗ U(ℂn), where U(ℂn) is the space of entirerapidly decreasing functions in any horizontal band of ℂn. We then use such continuity results to develop a framework for the STFT on K'1(ℝn). Also, we devote one section to the characterization of K’1(ℝn) and related spaces via modulation spaces. We also obtain various Tauberian theorems for the short-time Fourier transform. Part of the thesis is dedicated to the ridgelet and the Radon transform. We define and study the ridgelet transform of (Lizorkin) distributions and we show that the ridgelet transform and the ridgelet synthesis operator can be extended as continuous mappings Rψ : S’0(ℝn) → S’(Yn+1) and Rtψ: S’(Yn+1) → S’0(ℝn). We then use our results to develop a distrib...

We study the short-time Fourier transform on the space K ′ 1 (R n) of distributions of exponential type. We give characterizations of K ′ 1 (R n) and some of its subspaces in terms of modulation spaces. We also obtain various Tauberian theorems for the short-time Fourier transform.

Journal of Mathematical Analysis and Applications, 2002

A characterization of weighted L2(I) spaces in terms of their images under various integral transformations is derived, where I is an interval (finite or infinite). This characterization is then used to derive Paley-Wiener-type theorems for these spaces. Unlike the classical Paley-Wiener theorem, our theorems use real variable techniques and do not require analytic continuation to the complex plane. The class of integral transformations considered is related to singular Sturm-Liouville boundary-value problems on a half line and on the whole line. 1 1991 Mathematics Subject Classifications. Primary 44A15, 34B24; Secondary 42B10, 33C45

Journal of Fourier Analysis and Applications, 2010

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.

We study several Tauberian properties of regularizing transforms of tempered distributions with values in Banach spaces, 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). 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).

Random Operators and Stochastic Equations, 2009

For the Wiener integral, one property of inversion is established. This property is used for construction of nonparametric statistical estimation of the unknown logarithmic derivative for distribution random processes, which is observed in Wiener noise.

Analysis

We list multiplier and convolutor spaces of the spaces occurring in L. Schwartz' "Théorie des distributions". Furthermore we clarify whether the multiplications and convolutions are continuous or not.

Czechoslovak Mathematical Journal, 2013

We provide several general versions of Littlewood's Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We apply these Tauberian results to deduce a number of Tauberian theorems for power series where Cesàro summability follows from Abel summability. We also use our general results to give a new simple proof of the classical Littlewood one-sided Tauberian theorem for power series.

Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1983

We consider the determination of the behavior of a distribution function F at its endpoints in terms of the behavior of its Laplace-Stieltjes transform co at the limits of its interval of convergence. The results extend various known strong and weak results to a larger class of distributions via a relatively straightforward technique based on the weak convergence of suitably normalized associated distributions. An application and examples are considered briefly.

Archiv der Mathematik, 2008

We give a tauberian theorem for boundary values of analytic functions. We prove that if f ∈ D (a, b) is the distributional limit of the analytic function F defined in a region of the form (a, b) × (0, R) , if F (x0 + iy) → γ as y → 0 + , and if f is distributionally bounded at x = x0, then f (x0) = γ distributionally. As a consequence of our tauberian theorem, we obtain a new proof of a tauberian theorem of Hardy and Littlewood.