Tauberian Theorems Research Articles

An alternative proof of the generalized Littlewood Tauberian theorem for Cesàro summable double sequences

In this paper, we first examine the relationships between a double sequence and its arithmetic means in different senses (i. e. $(C,1,0)$, $(C,0,1)$ and $(C,1,1)$ means) in terms of slow oscillation in certain senses and investigate some properties of oscillatory behaviors of the difference sequence between the double sequence and its arithmetic means in different senses. Next, we give an alternative proof of the generalized Littlewood Tauberian theorem for Cesàro summability method as an application of the results obtained in the first part.

A bounds Tauberian theorem

A bounds Tauberian theorem

Abelian theorem for the regularly varying measure and its density in orthant

Рассматривается $\sigma$-конечная мера $U$, сосредоточенная в положительном координатном угле $\mathbf{R}^n_+=[0,\infty)^n$, для которой существует преобразование Лапласа $\widetilde{U}(\lambda)$ при $\lambda\in\operatorname{int} \mathbf{R}^n_+$. Пусть заданы функции $R(t)>0$ и $b(t)=(b_1(t),…,b_n(t))\in\operatorname{int} \mathbf{R}^n_+$ при $t\geq0$ такие, что $R(t)\to\infty$, $b_i(t)\to\infty$\enskip $\forall i=1,…,n$. При некоторых предположениях на эти функции, из слабой сходимости последовательности мер $U(b(t) {\cdot} )/R(t)$ к $\Phi{( \cdot )}$ при $t\to\infty$ выводится сходимость $\widetilde{U}(\lambda/b(t))\to\widetilde{\Phi}(\lambda)<\infty$ для любого $\lambda\in\operatorname{int} \mathbf{R}^n_+$ ($t\to\infty$) (умножение и деление векторов - покомпонентное). Функцию $f\colon \mathbf{R}_+^n\to \mathbf{R}_+$ назовем правильно меняющейся на бесконечности в $\mathbf{R}_+^n$ вдоль $b(t)$, если для всех $x$, $x(t) \in \mathbf{R}_+^n\setminus\{0\}\colon x(t)\to x$ выполнено соотношение $f(b(t)x(t))/f(b(t))\to\varphi(x)\in(0,\infty)$ при $t\to\infty$. Даны достаточные условия на такие функции, при выполнении которых $\widehat{f}(\lambda/b(t))\equiv\widetilde{U}(\lambda/b(t)) \to\widehat{\phi}(\lambda)\equiv\widetilde{\Phi}(\lambda)<\infty$ для любого $\lambda\in\operatorname{int} \mathbf{R}^n_+$\enskip ($t\to\infty$) при $U(dx)=f(x) dx$, $\Phi(dx)=\varphi(x) dx$. Полученная абелева теорема применяется в конце статьи для исследования предельного поведения распределения типа кратного степенного ряда.

Abelian Theorem for the Regularly Varying Measure and Its Density in Orthant

The paper is concerned with a $\sigma$-finite measure $U$ concentrated in the positive orthant $\mathbf{R}^n_+=[0,\infty)^n$ such that there exists the Laplace transform $\widetilde{U}(\lambda)$ for $\lambda\in\operatorname{int} \mathbf{R}^n_+$. Let functions $R(t)>0$ and $b(t)=(b_1(t),\dots,b_n(t))\in\operatorname{int} \mathbf{R}^n_+$ for $t\geq0$ be such that $R(t)\to\infty$, $b_i(t)\to\infty$ for any $i=1,\dots,n$. Under certain assumptions on these functions, the weak convergence of the measures $U(b(t)\,{\cdot}\,)/R(t)$ to $\Phi{(\,\cdot\,)}$ as $t\to\infty$ is shown to imply the convergence $\widetilde{U}(\lambda/b(t))\to\widetilde{\Phi}(\lambda)<\infty$ for any $\lambda\in\operatorname{int} \mathbf{R}^n_+$ ($t\to\infty$) (the multiplication and division of vectors are defined componentwise). A function $f\colon \mathbf{R}_+^n\to \mathbf{R}_+$ is said to be regularly varying at infinity in $\mathbf{R}_+^n$ along $b(t)$ if $f(b(t)x(t))/f(b(t))\to\varphi(x)\in(0,\infty)$ as $t\to\infty$ for all $x$, $x(t) \in \mathbf{R}_+^n\setminus\{0\}$ such that $ x(t)\to x$. Sufficient conditions are given for such functions to give $\widehat{f}(\lambda/b(t))\equiv\widetilde{U}(\lambda/b(t)) \to\widehat{\phi}(\lambda)\equiv\widetilde{\Phi}(\lambda)<\infty$ for any $\lambda\in\operatorname{int} \mathbf{R}^n_+$\enskip ($t\to\infty$) for $U(dx)=f(x)\,dx$, $\Phi(dx)=\varphi(x)\,dx$. The Abelian theorem obtained here is applied at the end of the paper to investigate the limit behavior of multiple power series distributions.

Unlimited parity alternating partitions

We introduce a new type of partition that consists of partitions whose different parts alternate in parity (e.g., 3 + 2 + 2 + 1 + 1). Various properties of this partition function are studied. In particular, we obtain its asymptotic behavior by employing Ingham’s Tauberian theorem.

Tauberian conditions on some slowly decreasing sequences

We present some Tauberian conditions to recover Cesaro summability of a sequence out of the product methods of Abel and Cesaro summability of the sequence. Moreover, we generalize some classical Tauberian theorems, such as the Hardy–Littlewood theorem, the generalized Littlewood theorem for Abel summability method.

On Tauberian theorem for stationary Nash equilibria

We consider general n-player nonzero-sum dynamic games, which is broader than differential games and could accommodate both discrete and continuous time. Assuming common dynamics, we study the long run average family and discounting average family of the running costs. For each of these game families, we investigate asymptotic properties of its Nash equilibria. We analyze asymptotic Nash equilibria—strategy profiles that are approximately optimal if the planning horizon tends to infinity in long run average games and if the discount tends to zero in discounting games. Moreover, we also assume that this strategy profile is stationary. Under a mild assumption on players’ strategy sets, we prove a uniform Tauberian theorem for stationary asymptotic Nash equilibrium. If a stationary strategy profile is an asymptotic Nash equilibrium and the corresponding Nash value functions converge uniformly for one of the families (when discount goes to zero for discounting games, when planning horizon goes to infinity in long run average games), then for the other family this strategy profile is also an asymptotic Nash equilibrium, and its Nash value functions converge uniformly to the same limit. As an example of application of this theorem, we consider Sorger’ model of competition of two firms.

Tauberian Conditions of Slowly Decreasing Type for the Logarithmic Power Series Method

Let \((s_n)\) be a sequence of real numbers. We say that \((s_n)\) is summable to \(\xi \) by the logarithmic power series method if $$\lim _{x\rightarrow 1^-}f(x)=\xi, \quad {\text{where}}\quad f(x)=-\frac{1}{\log (1-x)}\sum_{n=0}^{\infty }\frac{s_n}{n+1}x^{n+1}. $$ It is well known that if the limit \(\lim_{n \rightarrow \infty }s_n=\xi \) exists, then the limit $$\lim _{x \rightarrow 1^-} f(x)=\xi $$ also exists. In this paper, we determine Tauberian conditions of slowly decreasing type to obtain ordinary convergence of \((s_n)\) from its summability by logarithmic power series method. As a consequence of our result, we give a short proof of an earlier Tauberian theorem due to Kwee (Can J Math 20:1324–1331, 1968).

Remarks on the abelian convexity theorem

This paper contains some observations on abelian convexity theorems. Convexity along an orbit is established in a very general setting using Kempf–Ness functions. This is applied to give short proofs of the Atiyah–Guillemin–Sternberg theorem and of abelian convexity for the gradient map in the case of a real analytic submanifold of complex projective space. Finally we give an application to the action on the probability measures.

Hardy-Type Tauberian Conditions on Time Scales

Hardy’s well-known Tauberian theorem for number sequences states that if a sequence $$x=\left( x_{k}\right) $$ satisfies $$\lim Cx=L$$ and $$\Delta x_{k}:=x_{k+1}-x_{k}=O\left( 1/k\right) $$ , then $$\lim x=L$$ , where Cx denotes the Cesaro mean (arithmetic mean) of x. In this study, we extend this result to the theory of time scales. We also discuss the theory on some special time scales.

On the summability methods of logarithmic type and the Berezin symbol

We prove by means of the Berezin symbols some theorems for the $\left( L\right) $-summability method for sequences and series. Also, we prove a new Tauberian type theorem for $\left( L\right) $-summability.

Applications of Tauberian theorem in Orlicz spaces of double difference sequences of fuzzy numbers

Applications of Tauberian theorem in Orlicz spaces of double difference sequences of fuzzy numbers

Weighted Integrability and Its Applications in Quantum Calculus

The weighted method of integrability in quantum calculus is introduced in this paper. Its several properties and the q-analogues of some results in summability theory are developed. Furthermore, some classical type Tauberian theorems for the weighted integrability of infinite q-integrals are established.

NOTE ON THE NUMBER OF DIVISORS OF REDUCIBLE QUADRATIC POLYNOMIALS

Lapkova [‘On the average number of divisors of reducible quadratic polynomials’, J. Number Theory 180 (2017), 710–729] uses a Tauberian theorem to derive an asymptotic formula for the divisor sum $\sum _{n\leq x}d(n(n+v))$ where $v$ is a fixed integer and $d(n)$ denotes the number of divisors of $n$. We reprove this result with additional terms in the asymptotic formula, by investigating the relationship between this divisor sum and the well-known sum $\sum _{n\leq x}d(n)d(n+v)$.

Rates of convergence to equilibrium for collisionless kinetic equations in slab geometry

This work deals with free transport equations with partly diffuse stochastic boundary operators in slab geometry. Such equations are governed by stochastic semigroups in L1 spaces. We prove convergence to equilibrium at the rate O(t−k2(k+1)+1)(t→+∞) for L1 initial data g in a suitable subspace of the domain of the generator T where k∈N depends on the properties of the boundary operators near the tangential velocities to the slab. This result is derived from a quantified version of Ingham's tauberian theorem by showing that Fg(s):=limε→0+⁡(is+ε−T)−1g exists as a Ck function on R\\{0} such that ‖djdsjFg(s)‖≤C|s|2(j+1) near s=0 and bounded as |s|→∞(0≤j≤k). Various preliminary results of independent interest are given and some related open problems are pointed out.

Tauberian theorems for the Stockwell transform of Lizorkin distributions

ABSTRACTWe prove the continuity of the Stockwell transform and the corresponding synthesis operator on the spaces of highly localized test functions over and , respectively. Using the obtained continuity results, we define and study the Stockwell transform on the space of Lizorkin distributions . We also provide Abelian and Tauberian type results relating the asymptotic behavior of distributions with the asymptotics of their Stockwell transform.

Optimal Tauberian constant in Ingham’s theorem for Laplace transforms

It is well known that there is an absolute constant $\mathfrak{C}>0$ such that if the Laplace transform $G(s)=\int_{0}^{\infty}\rho(x)e^{-s x}\:\mathrm{d}x$ of a bounded function $\rho$ has analytic continuation through every point of the segment $(-i\lambda ,i\lambda )$ of the imaginary axis, then $$ \limsup_{x\to\infty} \left|\int_{0}^{x}\rho(u)\:\mathrm{d}u - G(0)\right|\leq \frac{ \mathfrak{C}}{\lambda} \: \limsup_{x\to\infty} |\rho(x)|. $$ The best known value of the constant $\mathfrak{C}$ was so far $\mathfrak{C}=2$. In this article we show that the inequality holds with $\mathfrak{C}=\pi/2$ and that this value is best possible. We also sharpen Tauberian constants in finite forms of other related complex Tauberian theorems for Laplace transforms.

Tauberian theorems for iterations of weighted mean summable integrals

Let p be a positive weight function on $$A:=[1, \infty )$$ which is integrable in Lebesgue’s sense over every finite interval (1, x) for $$1<x<\infty $$ , in symbol: $$p \in L^{1}_{loc} (A)$$ such that $$P(x)=\int _{1}^{x} p(t) dt\ne 0$$ for each $$x>1$$ , $$P(1)=0$$ and $$P(x) \rightarrow \infty $$ as $$x \rightarrow \infty $$ . For a real-valued function $$f \in L^{1}_{loc} (A)$$ , we set $$s(x):=\int _{1}^{x}f(t)dt$$ and denote $$\begin{aligned} \sigma ^{(0)}_p(x):=s(x), \sigma ^{(k)}_p(x):=\frac{1}{P(x)}\int _1^x \sigma ^{(k-1)}_p(t) p(t)dt\,\,\, (x>1, k=1,2,\ldots ), \end{aligned}$$ provided that $$P(x)>0$$ . If $$\begin{aligned} \lim _{x\rightarrow \infty }\sigma ^{(k)}_p(x)=L, \end{aligned}$$ we say that $$\int _{1}^{\infty }f(x)dx$$ is summable to L by the k-th iteration of weighted mean method determined by the function p(x), or for short, $$({\overline{N}},p,k)$$ integrable to a finite number L and we write $$s(x)\rightarrow L({\overline{N}},p,k)$$ . It is well-known that the existence of the limit $$\lim _{x \rightarrow \infty } s(x)=L$$ implies that of $$\lim _{x \rightarrow \infty } \sigma ^{(k)}_p(x)=L$$ . But the converse of this implication is not true in general. In this paper, we obtain some Tauberian theorems for the weighted mean method of integrals in order that the converse implication holds true. Our results extend and generalize some classical type Tauberian theorems given for Cesaro and logarithmic summability methods of integrals.

Tauberian theorems for the logarithmic summability methods of integrals

A real-valued continuous function f on $$[1, \infty )$$ is said to be summable by the logarithmic summability method of integrals (shortly, $$\ell $$ summable) if $$\begin{aligned} \lim _{x \rightarrow \infty }\frac{1}{\log {x}}\int _1^x\frac{s(t)}{t}dt \end{aligned}$$ exists, where $$s(x)=\int _{1}^{x}f(t)dt$$ . Our goal in this paper is to obtain some Tauberian theorems for the logarithmic summability method of integrals by some new Tauberian conditions. We also introduce the $$\ell ^{(k)}$$ summability method of integrals and give some Tauberian theorems for this method.

Tauberian theorems for statistical summability methods of sequences of fuzzy numbers

We give Tauberian conditions of slow decrease type under which statistical Cesaro summability and statistical logarithmic summability of sequences of fuzzy numbers imply convergence in fuzzy number space. Besides, we obtain a sufficient condition for statistically convergent sequences of fuzzy numbers to converge in the ordinary sense.