Tauberian Theorems Research Articles

Cesàro summability of sequences in intuitionistic fuzzy normed spaces and related Tauberian theorems

We define the concept of Cesaro summability method in intuitionistic fuzzy normed spaces and prove a related Tauberian theorem. Also, we define slowly oscillating sequences in intuitionistic fuzzy normed spaces, prove related theorems and show that Cesaro summability of slowly oscillating sequences implies ordinary convergence in intuitionistic fuzzy normed spaces. Finally, we give an analogue of classical two-sided Tauberian theorem due to Hardy by using the concept of q-boundedness.

An asymptotic analysis of the Fourier-Laplace transforms of certain oscillatory functions

We study the family of Fourier-Laplace transformsFα,β(z)=F.p.∫0∞tβexp⁡(itα−izt)dt,Imz<0, for α>1 and β∈C, where Hadamard finite part is used to regularize the integral when Reβ≤−1. We prove that each Fα,β has analytic continuation to the whole complex plane and determine its asymptotics along any line through the origin. We also apply our ideas to show that some of these functions provide concrete extremal examples for the Wiener-Ikehara theorem and a quantified version of the Ingham-Karamata theorem, supplying new simple and constructive proofs of optimality results for these complex Tauberian theorems.

Tauberian theorems for statistical logarithmic summability of strongly measurable fuzzy valued functions

We define statistical logarithmic summability of strongly measurable fuzzy valued functions and we give slowly decreasing type Tauberian conditions under which statistical limit at infinity and statistical logarithmic summability of strongly measurable fuzzy valued functions imply ordinary limit at infinity in one dimensional fuzzy number space $E^1$. Besides, we give slowly oscillating type Tauberian conditions for statistical limit and statistical logarithmic summability of strongly measurable fuzzy valued functions in $n-$dimensional fuzzy number space $E^n$.

Multiparameter Abelian ergodic theorems of Chacon–Báez–Duarte type

We prove a multiparameter Llogk L generalization of the Baez– Duarte Abelian ergodic theorem for positive linear contractions on L1, which allows the application of the local convergence principle of Sucheston’s type. Next we establish a new weighted Abelian ratio ergodic theorem for Dunford– Schwartz operators on L1 with modulation by Besicovitch sequences. Moreover, this (one-parameter) result is generalized to the case of multiparameter operator averages, which allows the application of Fava’s maximal ergodic inequality.

Quantum Harmonic Analysis on Lattices and Gabor Multipliers

We develop a theory of quantum harmonic analysis on lattices in {mathbb {R}}^{2d}. Convolutions of a sequence with an operator and of two operators are defined over a lattice, and using corresponding Fourier transforms of sequences and operators we develop a version of harmonic analysis for these objects. We prove analogues of results from classical harmonic analysis and the quantum harmonic analysis of Werner, including Tauberian theorems and a Wiener division lemma. Gabor multipliers from time-frequency analysis are described as convolutions in this setting. The quantum harmonic analysis is thus a conceptual framework for the study of Gabor multipliers, and several of the results include results on Gabor multipliers as special cases.

Bounds on the density of states and the spectral gap in CFT2

We improve the recently discovered upper and lower bounds on the $O(1)$ correction to the Cardy formula for the density of states integrated over an energy window (of width $2\delta$), centered at high energy in 2 dimensional conformal field theory. We prove optimality of the lower bound for $\delta\to 1^{-}$. We prove a conjectured upper bound on the asymptotic gap between two consecutive Virasoro primaries for a central charge greater than $1,$ demonstrating it to be $1.$ Furthermore, a systematic method is provided to establish a limit on how tight the bound on the $O(1)$ correction to the Cardy formula can be made using bandlimited functions. The techniques and the functions used here are of generic importance whenever the Tauberian theorems are used to estimate some physical quantities.

A note on cosine series with coefficients of generalized bounded variation

Abstract In this note, we obtain a Tauberian theorem for a class of regular lower triangular matrices operating on cosine series with coefficients tending to zero. As corollaries we obtain Tauberian theorems for weighted mean, Nörlund, and Hausdorff matrices.

Statistical Tauberian theorems for Ces\xe0ro integrability mean based on post-quantum calculus

The notion of statistical convergence is more general than the classical convergence. Tauberian theorems via different ordinary summability means have been established by many researchers. In the present work, we have established some new Tauberian theorems based on post-quantum calculus via statistical Cesàro summability mean of real-valued continuous function of one variable under oscillating behavior and De la vallée Poussin mean of a single integral. Moreover, some remarks and corollaries are provided here to support our theorems.

On a Problem of Multidimensional Tauberian Theory

In many Tauberian theorems, the asymptotic properties of functions were investigated with respect to a predefined function (usually in the scale of regularly varying functions). In this paper, we address an alternative problem: Given a generalized function, does it have asymptotics with respect to some regularly varying function? We find necessary and sufficient conditions for the existence of quasiasymptotics of those generalized functions whose Laplace transforms have a bounded argument in a tube domain over the positive orthant. Moreover, we point out a regularly varying function with respect to which quasiasymptotics exists. It turns out that the modulus of a holomorphic function in a tube domain over the positive orthant in the purely imaginary subspace on rays emanating from the origin behaves as a regularly varying function. We use the obtained results to find the quasiasymptotics of the generalized Cauchy problem for convolution equations whose kernels are passive operators.

Tauberian theorems for k–summability with respect to an analytic germ

The goal of this article is to establish Tauberian theorems for the k–summability processes defined by germs of analytic functions in several complex variables. The proofs are based on the Tauberian theorems for k–summability in one variable and in monomials, and a method of monomialization of germs of analytic functions.

A Tauberian Theorem for a Weighted Mean Method of Summability in Ordered Spaces

In this paper, we extend a Tauberian theorem for the Cesaro summability method due to Maddox (Analysis 9:297–302, 1989) in ordered spaces to the weighted mean method of summability.

An alternative proof of a Tauberian theorem for the weighted mean summability of integrals over R_{+}

An alternative proof of a Tauberian theorem for the weighted mean summability of integrals over R_{+}

Regularly log-periodic functions and some applications

We prove a Tauberian theorem for the Laplace--Stieltjes transform and Karamata-type theorems in the framework of regularly log-periodic functions. As an application we determine the exact tail behavior of fixed points of certain type smoothing transforms.

Tauberian-Cardy formula with spin

We prove a 2 dimensional Tauberian theorem in context of 2 dimensional conformal field theory. The asymptotic density of states with conformal weight (h, overline{h} ) → (∞, ∞) for any arbitrary spin is derived using the theorem. We further rigorously show that the error term is controlled by the twist parameter and insensitive to spin. The sensitivity of the leading piece towards spin is discussed. We identify a universal piece in microcanonical entropy when the averaging window is large. An asymptotic spectral gap on (h, overline{h} ) plane, hence the asymptotic twist gap is derived. We prove an universal inequality stating that in a compact unitary 2D CFT without any conserved current Agle frac{pi left(c-1right){r}^2}{24} is satisfied, where g is the twist gap over vacuum and A is the minimal “areal gap”, generalizing the minimal gap in dimension to (h′, overline{h}^{prime } ) plane and r=frac{4sqrt{3}}{pi}simeq 2.21 . We investigate density of states in the regime where spin is parametrically larger than twist with both going to infinity. Moreover, the large central charge regime is studied. We also probe finite twist, large spin behavior of density of states.

Bound on asymptotics of magnitude of three point coefficients in 2D CFT

We use methods inspired from complex Tauberian theorems to make progress in understanding the asymptotic behavior of the magnitude of heavy-light-heavy three point coefficients rigorously. The conditions and the precise sense of averaging, which can lead to exponential suppression of such coefficients are investigated. We derive various bounds for the typical average value of the magnitude of heavy-light-heavy three point coefficients and verify them numerically.

Convergence follows from Cesàro summability in the case of slowly decreasing or slowly oscillating double sequences in certain senses

Let (u??) be a double sequence of real or complex numbers which is (C; 1; 1) summable to a finite limit. We obtain some Tauberian conditions of slow decreasing or oscillating types in terms of the generator sequences in certain senses under which P-convergence of a double sequence (u??) follows from its (C,1,1) summability. We give Tauberian theorems in which Tauberian conditions are of Hardy and Landau types as special cases of our results. We present some Tauberian conditions in terms of the de la Vall?e Poussin means of double sequences under which P-convergence of a double sequence (u??) follows from its (C,1,1) summability. Moreover, we give analogous results for (C,1,0) and (C,0,1) summability methods.

Abelian theorems for Laplace and Mehler-Fock transforms of generalized functions

The main purpose of this article is to exhibit Abelian theorems for the Laplace and the Mehler-Fock transforms of general order over distributions of compact support and over certain spaces of generalized functions.

A multidimensional Tauberian theorem for Laplace transforms of ultradistributions

ABSTRACTWe obtain a multidimensional Tauberian theorem for Laplace transforms of Gelfand-Shilov ultradistributions. The result is derived from a Laplace transform characterization of bounded sets in spaces of ultradistributions with supports in a convex acute cone of , also established here.

Asymptotic and Non-asymptotic Estimates for Multivariate Laplace

We derive bilateral asymptotic as well as non-asymptotic estimates for the multivariate Laplace integrals. Furthermore, we provide multidimensional Tauberian theorems for exponential integrals.

A Tauberian theorem for the statistical generalized Nörlund-Euler summability method

Abstract Let (pn) and (qn) be any two non-negative real sequences with R n : = ∑ k = 0 n p k q n - k ≠ 0 ( n ∈ 𝕅 ) {{\rm{R}}_{\rm{n}}}: = \sum\limits_{{\rm{k}} = 0}^{\rm{n}} {{{\rm{p}}_{\rm{k}}}{{\rm{q}}_{{\rm{n}} - {\rm{k}}}}} \ne 0\,\,\,\,\left( {{\rm{n}} \in {\rm\mathbb{N}}} \right) With E n 1 {\rm{E}}_{\rm{n}}^1 − we will denote the Euler summability method. Let (xn) be a sequence of real or complex numbers and set N p , q n E n 1 : = 1 R n ∑ k = 0 n p k q n - k 1 2 k ∑ v = 0 k ( v k ) x v {\rm{N}}_{{\rm{p}},{\rm{q}}}^{\rm{n}}{\rm{E}}_{\rm{n}}^1: = {1 \over {{{\rm{R}}_{\rm{n}}}}}\sum\limits_{{\rm{k}} = 0}^{\rm{n}} {{{\rm{p}}_{\rm{k}}}{{\rm{q}}_{{\rm{n - k}}}}{1 \over {{2^{\rm{k}}}}}\sum\limits_{{\rm{v}} = 0}^{\rm{k}} {\left( {_{\rm{v}}^{\rm{k}}} \right){{\rm{x}}_{\rm{v}}}} } for n ∈ ℕ. In this paper, we present necessary and sufficient conditions under which the existence of the st− limit of (xn) follows from that of st - N p , q n E n 1 {\rm{st - N}}_{{\rm{p}},q}^{\rm{n}}{\rm{E}}_{\rm{n}}^1 − limit of (xn). These conditions are one-sided or two-sided if (xn) is a sequence of real or complex numbers, respectively.