Tauberian Theorems Research Articles

Tauberian conditions under which convergence follows from summability by thediscrete power series method

In this paper, we obtain Tauberian conditions to recover convergence of a series from its discrete power series summability under certain conditions. As special cases of our main results, we get discrete analogues of some well-known Tauberian theorems in the literature.

A Tauberian theorem for ideal statistical convergence

Given an ideal I on the positive integers, a real sequence (xn) is said to be I-statistically convergent to ℓ provided that n∈N:1n|{k≤n:xk∉U}|≥ε∈Ifor all neighborhoods U of ℓ and all ε>0. First, we show that I-statistical convergence coincides with J-convergence, for some unique ideal J=J(I). In addition, J is Borel [analytic, coanalytic, respectively] whenever I is Borel [analytic, coanalytic, resp.].Then we prove, among others, that if I is the summable ideal {A⊆N:∑a∈A1∕a<∞} or the density zero ideal {A⊆N:limn→∞1n|A∩[1,n]|=0} then I-statistical convergence coincides with statistical convergence. This can be seen as a Tauberian theorem which extends a classical theorem of Fridy. Lastly, we show that this is never the case if I is maximal.

Tauberian Theorems for the Product of Weighted and Cesàro Summability Methods for Double Sequences

In this paper, we obtain necessary and sufficient conditions, under which convergence of a double sequence in Pringsheim's sense follows from its weighted-Cesaro summability. These Tauberian conditions are one-sided or two-sided if it is a sequence of real or complex numbers, respectively.

Modular invariance, tauberian theorems and microcanonical entropy

We analyze modular invariance drawing inspiration from tauberian theorems. Given a modular invariant partition function with a positive spectral density, we derive lower and upper bounds on the number of operators within a given energy interval. They are most revealing at high energies. In this limit we rigorously derive the Cardy formula for the microcanonical entropy together with optimal error estimates for various widths of the averaging energy shell. We identify a new universal contribution to the microcanonical entropy controlled by the central charge and the width of the shell. We derive an upper bound on the spacings between Virasoro primaries. Analogous results are obtained in holographic 2d CFTs. We also study partition functions with a UV cutoff. Control over error estimates allows us to probe operators beyond the unity in the modularity condition. We check our results in the 2d Ising model and the Monster CFT and find perfect agreement.

Analytic Euclidean bootstrap

We solve crossing equations analytically in the deep Euclidean regime. Large scaling dimension ∆ tails of the weighted spectral density of primary operators of given spin in one channel are matched to the Euclidean OPE data in the other channel. Subleading frac{1}{varDelta } tails are systematically captured by including more operators in the Euclidean OPE in the dual channel. We use dispersion relations for conformal partial waves in the complex ∆ plane, the Lorentzian inversion formula and complex tauberian theorems to derive this result. We check our formulas in a few examples (for CFTs and scattering amplitudes) and find perfect agreement. Moreover, in these examples we observe that the large ∆ expansion works very well already for small ∆ ∼ 1. We make predictions for the 3d Ising model. Our analysis of dispersion relations via complex tauberian theorems is very general and could be useful in many other contexts.

Many-dimensional tauberian theorems for holomorphic functions of bounded argument

For generalized functions with Laplace transform has nonnegative imaginary part in tube domain over positive actant, we found sufficient conditions for existence of quasiasymptotic, the function with regular behavior with respect to which the quasiasymptotic exists being explicitly found. The obtained results are used to steady of the asymptotic behaviour of solutions of the Cauchy problem of passive operators.

An abstract approach to optimal decay of functions and operator semigroups

We provide a new and significantly shorter optimality proof of recent quantified Tauberian theorems, both in the setting of vector-valued functions and of $C_0$-semigroups, and in fact our results are also more general than those currently available in the literature. Our approach relies on a novel application of the open mapping theorem.

Slower variation of the generation sizes induced by heavy-tailed environment for geometric branching

Motivated by seminal paper of Kozlov et al. Kesten et al. (1975) we consider in this paper a branching process with a geometric offspring distribution parametrized by random success probability A and immigration equals 1 in each generation. In contrast to above mentioned article, we assume that environment is heavy-tailed, that is logA−1(1−A) is regularly varying with a parameter α>1, that is that P(logA−1(1−A)>x)=x−αL(x) for a slowly varying function L. We will prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of distribution of the population at nth generation which gets even heavier with n increasing. Precisely, in this work, we prove that asymptotic tail P(Zl≥m) of lth population Zl is of order (log(l)m)−αL(log(l)m) for large m, where log(l)m=log…logm. The proof is mainly based on Tauberian theorem. Using this result we also analyze the asymptotic behavior of the first passage time Tn of the state n∈Z by the walker in a neighborhood random walk in random environment created by independent copies (Ai:i∈Z) of (0,1)-valued random variable A.

Some Tauberian theorems for weighted means of double integrals

Let p(x) and q(y) be nondecreasing continuous functions on [0,∞) such that p(0)=q(0)=0 and p(x),q(y)→∞ as x,y→∞. For a locally integrable function f(x,y) on R₊²=[0,∞)×[0,∞), we denote its double integral by F(x,y)=∫₀^{x}∫₀^{y}f(t,s)dtds and its weighted mean of type (α,β) by t_{α,β}(x,y)=∫₀^{x}∫₀^{y}(1-((p(t))/(p(x))))^{α}(1-((q(s))/(q(y))))^{β}f(t,s)dtds where α>-1 and β>-1. We say that ∫₀^{∞}∫₀^{∞}f(t,s)dtds is integrable to L by the weighted mean method of type (α,β) determined by the functions p(x) and q(x) if lim_{x,y→∞}t_{α,β}(x,y)=L exists. We prove that if lim_{x,y→∞}t_{α,β}(x,y)=L exists and t_{α,β}(x,y) is bounded on R₊² for some α>-1 and β>-1, then lim_{x,y→∞}t_{α+h,β+k}(x,y)=L exists for all h>0 and k>0. Finally, we prove that if ∫₀^{∞}∫₀^{∞}f(t,s)dtds is integrable to L by the weighted mean method of type (1,1) determined by the functions p(x) and q(x) and conditions [displaystyle] \displaystyle ((p(x))/(p′(x)))∫₀^{y}f(x,s)ds=O(1) and [displaystyle] \displaystyle ((q(y))/(q′(y)))∫₀^{x}f(t,y)dt=O(1) hold, then lim_{x,y→∞}F(x,y)=L exists.

Complex Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior

We provide several Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior. Our results generalize and improve various known versions of the Ingham-Fatou-Riesz theorem and the Wiener-Ikehara theorem. Using local pseudofunction boundary behavior enables us to relax boundary requirements to a minimum. Furthermore, we allow possible null sets of boundary singularities and remove unnecessary uniformity conditions occurring in earlier works; to this end, we obtain a useful characterization of local pseudofunctions. Most of our results are proved under one-sided Tauberian hypotheses; in this context, we also establish new boundedness theorems for Laplace transforms with pseudomeasure boundary behavior. As an application, we refine various results related to the Katznelson-Tzafriri theorem for power series.

Abelian theorems for distributional Kontorovich–Lebedev and Mehler–Fock transforms of general order

Our goal in this article is to derive Abelian theorems for the Kontorovich–Lebedev and Mehler–Fock transforms of general order over distributions of compact support and over certain spaces of generalized functions.

Abel extensions of some classical Tauberian theorems

The well-known classical Tauberian theorems given for Aλ (the discrete Abel mean) by Armitage and Maddox in [Armitage, H. D and Maddox, J. I., Discrete Abel means, Analysis, 10 (1990), 177–186] is generalized. Similarly the ”one-sided” Tauberian theorems of Landau and Schmidt for the Abel method are extended by replacing lim As with Abel-lim Aσi n(s). Slowly oscillating of {sn} is a Tauberian condition of the Hardy-Littlewood Tauberian theorem for Borel summability which is also given by replacing limt(Bs)t = `, where t is a continuous parameter, with limn(Bs)n = `, and further replacing it by Abel-lim(Bσi k (s))n = `, where B is the Borel matrix method.

On Wiener’s Tauberian theorems and convolution for oscillatory integral operators

The main aim of this work is to obtain Paley--Wiener and Wiener's Tauberian results associated with an oscillatory integral operator, which depends on cosine and sine kernels, as well as to introduce a consequent new convolution. Additionally, a new Young-type inequality for the obtained convolution is proven, and a new Wiener-type algebra is also associated with this convolution.

Tauberian theorems for $$(\overline{N},p,q)$$ summable double sequences of fuzzy numbers

In this paper, we define the weighted mean method $$(\overline{N},p,q)$$ of double sequences of fuzzy numbers and give necessary and sufficient Tauberian conditions under which convergence in Pringsheim’s sense of a double sequence of fuzzy numbers follows from its $$(\overline{N},p,q)$$ summability. These conditions are weaker than the weighted analogues of Landau’s conditions and Schmidt’s slow oscillation condition in some senses for two-dimensional case.

Tauberian Theorem for Games with Unbounded Running Cost

We study two game families with total payoffs that are defined either as the Cesaro average (the long run average game family) or Abel average (the discounting game family) of the running costs. We study value functions for all sufficiently small discounts and for all sufficiently large planning horizons (asymptotic approach), investigate a robust strategy that provides a near-optimal total payoff in this case (uniform approach). Assuming the Dynamic Programming Principle, we prove the corresponding Tauberian theorems without requiring the boundedness of the running cost.

Logarithmic means of sequences of fuzzy numbers and a Tauberian theorem

A sequence $$(x_n)$$ of fuzzy numbers is said to be summable to a fuzzy number L by the logarithmic mean method $$(\ell ,2)$$ if $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{\ell _n^{(2)}}\sum _{k=1}^{n}\frac{x_k}{k\ell _k}=L \end{aligned}$$where $$\begin{aligned} \ell _n^{(2)}=\sum _{k=1}^{n}\frac{1}{k\ell _k}\sim \log (\log n). \end{aligned}$$We prove that the ordinary convergence of $$(x_n)$$ implies its $$(\ell ,2)$$ summability. The converse implication is not necessarily true. Namely, the $$(\ell ,2)$$ summability of $$(x_n)$$ may not imply the convergence of $$(x_n)$$. However, under certain additional conditions the converse may hold. Such conditions are called Tauberian conditions, and the resulting theorem is said to be a Tauberian theorem. In this paper, we provide necessary and sufficient Tauberian conditions to transform $$(\ell ,2)$$ summable sequences of fuzzy numbers into convergent sequences of fuzzy numbers with preserving the limit.

Some Tauberian theorems for iterations of Hölder integrability method

Let f be a real or complex-valued function on $$[1,\infty )$$ which is continuous over every finite interval [1, x) for $$1<x<\infty $$ . We set $$s(x):=\int _{1}^{x}f(t)dt$$ and define $$\sigma _{k}(s(x))$$ by $$\begin{aligned} \sigma _{k}(s(x))=\left\{ \begin{array}{ll} \displaystyle {\frac{1}{x}\int _{1}^{x}} \sigma _{k-1}(s(t))dt,&{}\quad k\ge 1\\ s(x),&{}\quad k=0 \end{array} \right. \end{aligned}$$ for each nonnegative integer k. An improper integral $$\begin{aligned} \int _{1}^{\infty } f(x)dx \end{aligned}$$ is said to be integrable to a finite number $$\mu $$ by the k-th iteration of Holder or Cesaro mean method of order one, or for short, the (H, k) integrable to $$\mu $$ if $$\begin{aligned} \lim _{x\rightarrow \infty }\sigma _{k}(s(x))=\mu . \end{aligned}$$ In this case, we write $$s(x)\rightarrow \mu \,\,(H,k)$$ . It is clear that the (H, k) integrability method reduces to the ordinary convergence for $$k=0$$ and the (H, 1) integrability method is (C, 1) integrability method. It is known that $$\lim _{x \rightarrow \infty } s(x) =\mu $$ implies $$\lim _{x \rightarrow \infty }\sigma _{k}(s(x)) =\mu $$ . But the converse of this implication is not true in general. In this paper, we obtain some Tauberian conditions for the iterations of Holder integrability method under which the converse implication holds.

Mixed-State Localization Operators: Cohen’s Class and Trace Class Operators

We study mixed-state localization operators from the perspective of Werner’s operator convolutions which allows us to extend known results from the rank-one case to trace class operators. The idea of localizing a signal to a domain in phase space is approached from various directions such as bounds on the spreading function, probability densities associated to mixed-state localization operators, positive operator valued measures, positive correspondence rules and variants of Tauberian theorems for operator translates. Our results include a rigorous treatment of multiwindow-STFT filters and a characterization of mixed-state localization operators as positive correspondence rules. Furthermore we provide a description of the Cohen class in terms of Werner’s convolution of operators and deduce consequences on positive Cohen class distributions, an uncertainty principle, uniqueness and phase retrieval for general elements of Cohen’s class.

An Alternative Proof of a Tauberian Theorem for the Weighted Mean Method of Summability

Let $$(p_n)$$ be a sequence of nonnegative numbers such that $$p_0>0$$ and $$\begin{aligned} P_n:=\sum _{k=0}^{n}p_k\rightarrow \infty, n\rightarrow \infty . \end{aligned}$$ Let $$(u_n)$$ be a sequence of real or complex numbers. The nth weighted mean of $$(u_n)$$ is defined by $$\begin{aligned} \sigma _n:=\frac{1}{P_n}\sum _{k=0}^{n}p_k u_k \quad (n =0,1,2,\ldots ) \end{aligned}$$ We give an alternative proof of a Tauberian theorem stating that the existence of the limit $$\lim _{n \rightarrow \infty } u_n=s$$ follows from that of $$\lim _{n \rightarrow \infty } \sigma _n=s$$ and a Tauberian condition. If $$(u_n)$$ is a sequence of real numbers, then these Tauberian conditions are one-sided. If $$(u_n)$$ is a sequence of complex numbers, these Tauberian conditions are two-sided. Significance Statement: If a sequence converges, then its weighted means converge to the same number. But, the converse of this implication is not true in general and its partial converse might be valid. This manuscript presents an alternative proof of a well-known Tauberian theorem stating that convergence of a slowly decreasing sequence (in case of sequences of real numbers) or a slowly oscillating sequence (in case of sequences of complex numbers) follows from its weighted mean summability. Corollaries of the main results in this manuscript consist of well-known Tauberian theorems for Cesaro and logarithmic summability methods.

Tauberian class estimates for vector-valued distributions

We study Tauberian properties of regularizing transforms of vector-valued tempered distributions. The transforms have the form , where the kernel is a test function and . We investigate conditions which ensure that a distribution that a priori takes values in a 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 . Our results generalize and improve earlier Tauberian theorems due to Drozhzhinov and Zav’yalov. Special attention is paid to finding the optimal class of kernels for which these Tauberian results hold. Bibliography: 24 titles.