Tauberian Theorems Research Articles

Tauberian theorems on ℝ⁺ and applications

Let f f be a bounded and uniformly continuous function from R \mathbb {R} to a Banach space X X and s p ( f ) \mathbf {sp}(f) be its Carleman spectrum. A classical Tauberian theorem states that f f is constant if and only if s p ( f ) ⊂ { 0 } \mathbf {sp}(f) \subset \{ 0 \} , and f f is ω \omega -periodic if and only if s p ( f ) ⊂ 2 π ω Z \mathbf {sp}(f) \subset \frac {2\pi }{\omega } \mathbb {Z} for some ω > 0 \omega >0 . However, one cannot expect analogous results on R + \mathbb {R}^+ since there is a counterexample showing that the case of R + \mathbb {R}^+ contrasts dramatically with the case of R \mathbb {R} . In this paper, we succeed in extending the above classical Tauberian theorem to R + \mathbb {R}^+ and obtain an extension of the well-known Ingham theorem. We also apply our Tauberian theorems to abstract Cauchy problems and improve a result in [Russian Math. 58 (2014), pp. 1–10]. Moreover, as an application, we present an extension of a Katznelson-Tzafriri theorem in [J. Funct. Anal. 103 (1992), pp. 74–84] with weaker assumptions. In addition, it is interesting to note that several of our results and examples show that S \mathcal {S} -asymptotically ω \omega -periodic functions on R + \mathbb {R}^{+} is just the “natural” analogue of periodic functions on R \mathbb {R} .

Sum rules & Tauberian theorems at finite temperature

We study CFTs at finite temperature and derive explicit sum rules for one-point functions of operators by imposing the KMS condition and we explicitly estimate one-point functions for light operators. Turning to heavy operators we employ Tauberian theorems and compute the asymptotic OPE density for heavy operators, from which we extract the leading terms of the OPE coefficients associated with heavy operators. Furthermore, we approximate and establish bounds for the two-point functions.

Statistical Cesàro Summability in Intuitionistic Fuzzy n-Normed Linear Spaces Leading towards Tauberian Theorems

Statistical Cesàro Summability in Intuitionistic Fuzzy n-Normed Linear Spaces Leading towards Tauberian Theorems

Tail asymptotics and precise large deviations for some Poisson cluster processes

Abstract We study the tail asymptotics of two functionals (the maximum and the sum of the marks) of a generic cluster in two sub-models of the marked Poisson cluster process, namely the renewal Poisson cluster process and the Hawkes process. Under the hypothesis that the governing components of the processes are regularly varying, we extend results due to [6, 19], notably relying on Karamata’s Tauberian Theorem to do so. We use these asymptotics to derive precise large-deviation results in the fashion of [32] for the just-mentioned processes.

Abelian Theorems for the Wavelet Transform in Terms of the Fractional Hankel Transform

Abelian Theorems for the Wavelet Transform in Terms of the Fractional Hankel Transform

A simple proof of the Wiener–Ikehara Tauberian Theorem

The Wiener–Ikehara Tauberian theorem is an important theorem giving an asymptotic formula for the sum of coefficients of a Dirichlet series ∑n=1∞a(n)ns. We provide a simple and elegant proof of the Wiener–Ikehara Tauberian theorem which relies only on basic Fourier analysis and known estimates for the given Dirichlet series. This method also allows us to derive a version of the Wiener–Ikehara theorem with an error term.

On weighted generator of triple sequences and its Tauberian conditions

This paper presents a new perspective on the relationship between the (N, p, q, r) method and P-convergence for triple sequences.Our primary goal is to derive Tauberian conditions that dictate the behavior of the weighted generator sequence (zlmn) in relation to the sequences (Pl), (Qm), and (Rn), with the intention of establishing a fresh interpretation. These conditions control the OL- and O-oscillatory properties and establish a connection from ( N, p, q, r) summability to P-convergence, subject to certain restrictions on the weight sequences. Furthermore, we demonstrate that specific cases, such as the OL-condition of Landau type and the O-condition of Hardy type with respect to (Pl), (Qm) and (Rn), serve as Tauberian conditions for (N, p, q, r) summability under additional conditions. Consequently, our findings encompass all the classical Tauberian theorems, including conditions related to slow decrease and slow oscillation in specific contexts.

Some Tauberian Theorems for the Weighted Mean Method of Summability of Double Sequences

Some Tauberian Theorems for the Weighted Mean Method of Summability of Double Sequences

Karamata's theorem for regularized Cauchy transforms

We prove Abelian and Tauberian theorems for regularized Cauchy transforms of positive Borel measures on the real line whose distribution functions grow at most polynomially at infinity. In particular, we relate the asymptotics of the distribution functions to the asymptotics of the regularized Cauchy transform.

New Tauberian Theorems for Cesáro Summable Triple Sequences of Fuzzy Numbers

The purpose of this paper is to establish new results on Tauberian theorem for Cesàro summability of triple sequences of fuzzy numbers. Besides, we extend and unify several results in the available literature. Furthermore, a huge number of special cases, theorems and their implications are proved. We show some illustrative examples in support of the results obtained in this paper.

Wiener Tauberian theorem and half-space problems for parabolic and elliptic equations

<abstract><p>For various kinds of parabolic and elliptic partial differential and differential-difference equations, results on the stabilization of solutions are presented. For the Cauchy problem for parabolic equations, the stabilization is treated as the existence of a limit as the time unboundedly increases. For the half-space Dirichlet problem for parabolic equations, the stabilization is treated as the existence of a limit as the independent variable orthogonal to the boundary half-plane unboundedly increases. In the classical case of the heat equation, the necessary and sufficient condition of the stabilization consists of the existence of the limit of mean values of the initial-value (boundary-value) function over balls as the ball radius tends to infinity. For all linear problems considered in the present paper, this property is preserved (including elliptic equations and differential-difference equations). The Wiener Tauberian theorem is used to establish this property. To investigate the differential-difference case, we use the fact that translation operators are Fourier multipliers (as well as differential operators), which allows one to use a standard Gel'fand-Shilov operational scheme. For all quasilinear problems considered in the present paper, the mean value from the stabilization criterion is changed: It undergoes a monotonic map, which is explicitly constructed for each investigated nonlinear boundary-value problem.</p></abstract>

General logarithmic control modulo and Tauberian remainder theorems

Let $\lambda=(\lambda_n)$ be a nondecreasing sequence of positive numbers such that $\lambda_n\to\infty$. A sequence $(\xi_n)$ is called $\lambda$-bounded if \begin{equation*} \lambda_n(\xi_n-\alpha)=O(1)\end{equation*} with the limit $\displaystyle{\lim_{n\rightarrow \infty}\xi_n=\alpha}$. In this work, we obtain several Tauberian remainder theorems on $\lambda$-bounded sequences for the logarithmic summability method with help of general logarithmic control modulo of the oscillatory behavior. Tauber conditions in our main results are on the generator sequence and the general logarithmic control modulo.

A new proof of the Erdos-Kac central limit theorem

In this paper we use the Riemann zeta distribution to give a new proof of the Erdös-Kac Central Limit Theorem. That is, if ζ ( s ) = ∑ n ≥ 1 1 n s \zeta (s)=\sum _{n\ge 1} \frac {1}{n^s} , s > 1 , s>1, then we consider the random variable X s X_s with P ( X s = n ) = 1 ζ ( s ) n s , P(X_s=n)=\frac {1}{\zeta (s)n^s}, n ≥ 1. n\ge 1. In an earlier paper, the first author and Adrien Peltzer derived the analog of the Erdös-Kac Central Limit Theorem (CLT) for the number of distinct prime factors, ω ( X s ) , \omega (X_s), of X s , X_s, as s ↘ 1. s\searrow 1. In this paper we show, by means of a Tauberian Theorem, how to obtain the Central Limit Theorem of Erdös-Kac for the uniform distribution from the result for the random variable X s X_s . We also apply the technique to the number of distinct prime divisors of X s X_s that lie in an arithmetic sequence and a local CLT of the type proved by Dixit and Murty [Hardy-Ramanujan J. 43 (2020), 17–23] as well a version of the CLT for irreducible divisors of a monic polynomial over a finite field.

A quantitative formula for the imaginary part of a Weyl coefficient

We investigate two-dimensional canonical systems y'=zJHy on an interval, with positive semi-definite Hamiltonian H . Let q_H be the Weyl coefficient of the system. We prove a formula that determines the imaginary part of q_H along the imaginary axis up to multiplicative constants, which are independent of H . We also provide versions of this result for Sturm–Liouville operators and Krein strings. Using classical Abelian–Tauberian theorems, we deduce characterizations of spectral properties such as integrability of a given comparison function with respect to the spectral measure \mu_H , and boundedness of the distribution function of \mu_H relative to a given comparison function. We study in depth Hamiltonians for which \arg q_H(ir) approaches 0 or \pi (at least on a subsequence). It turns out that this behavior of q_H(ir) imposes a substantial restriction on the growth of |q_H(ir)| . Our results in this context are interesting also from a function theoretic point of view.

Some Tauberian theorems for the weighted mean method of summability of double sequence

UDC 517.5 Let p = ( p j ) and q = ( q k ) be real sequences of nonnegative numbers with the property that P m = ∑ j = 0 m p j ≠ 0 and Q n = ∑ k = 0 n q k ≠ 0 for all m and n . Let ( P m ) and ( Q n ) be regulary varying positive indices. Assume that ( u m n ) is a double sequence of complex (real) numbers, which is ( N ¯ , p , q ; α , β ) summable with a finite limit, where ( α , β ) = ( 1,1 ) , ( 1,0 ) , or ( 0,1 ) . We present some conditions imposed on the weights under which ( u m n ) converges in Pringsheim's sense. These results generalize and extend the results obtained by authors in [Comput. Math. Appl., <strong>62</strong>, No. 6, 2609–2615 (2011)].

WIENER TAUBERIAN THEOREMS FOR CERTAIN BANACH ALGEBRAS ON REAL RANK ONE SEMISIMPLE LIE GROUPS

AbstractWe prove Wiener Tauberian theorem type results for various spaces of radial functions, which are Banach algebras on a real-rank-one semisimple Lie group G. These are natural generalizations of the Wiener Tauberian theorem for the commutative Banach algebra of the integrable radial functions on G.

On categories 𝒪 of quiver varieties overlying the bouquet graphs

We study representation theory of quantizations of Nakajima quiver varieties associated to bouquet quivers. We show that there are no finite dimensional representations of the quantizations A ¯ λ ( n , ℓ ) \overline {\mathcal {A}}_{\lambda }(n, \ell ) if both dim ⁡ V = n \operatorname {dim}V=n and the number of loops ℓ \ell are greater than 1 1 . We show that when n ≤ 3 n\leq 3 there is a Hamiltonian torus action with finitely many fixed points, provide the dimensions of Hom-spaces between standard objects in category O \mathcal {O} and compute the multiplicities of simples in standards for n = 2 n=2 in case of one-dimensional framing and generic one-parameter subgroups. We establish the abelian localization theorem and find the values of parameters, for which the quantizations have infinite homological dimension.

Tauberian theorems concerning weighted mean summable integrals

Tauberian theorems concerning weighted mean summable integrals

Tauberian theorems for smooth functions that never vanish

It is well known that if a function has slow growth in the ordinary sense, then it has slow growth in the Cesàro sense. The converse holds if special extra conditions are assumed; results of this form are called Tauberian Theorems. In this paper, we use techniques from generalized functions to investigate this phenomenon, with particular emphasis on the space of multipliers of Schwartz functions. We then apply our results to study series of delta functions and the division problem for tempered distributions.

Polynomial approximations in a generalized Nyman–Beurling criterion

The Nyman-Beurling criterion, equivalent to the Riemann hypothesis, is an approximation problem in the space of square integrable functions on $(0,\infty)$, involving dilations of the fractional part function by factors $\theta_k\in(0,1)$, $k\ge1$. Randomizing the $\theta_k$ generates new structures and criteria. One of them is a sufficient condition that splits into (i) showing that the indicator function can be approximated by convolution with the fractional part, (ii) a control on the coefficients of the approximation. This self-contained paper aims at identifying functions for which (i) holds unconditionally, by means of polynomial approximations. This yields in passing a short probabilistic proof of a known consequence of Wiener's Tauberian theorem. In order to tackle (ii) in the future, we give some expressions of the scalar products. New and remarkable structures arise for the Gram matrix, in particular moment matrices for a suitable weight that may be the squared $\Xi$-function for instance.