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.

For an infinite Toeplitz matrix T with nonnegative real entries we find the conditions under which the equation $\boldsymbol {x}=T\boldsymbol {x}$ , where $\boldsymbol {x}$ is an infinite vector column, has a nontrivial bounded positive solution. The problem studied in this paper is associated with the asymptotic behaviour of convolution-type recurrence relations and can be applied to problems arising in the theory of stochastic processes and other areas.

We investigate the spectral theory of integrable actions of a locally compact abelian group on a locally convex vector space. We start with an analysis of various spectral subspaces induced by the action of the group. This is applied to analyse the spectral theory of operators on the space, generated by measures on the group. We apply these results to derive general Tauberian theorems that apply to arbitrary locally compact abelian groups acting on a large class of locally convex vector spaces, which includes Fréchet spaces. We show how these theorems simplify the derivation of Mean Ergodic theorems.

PageRank with personalization is used in Web search as an importance measure for Web documents. The goal of this paper is to characterize the tail behavior of the PageRank distribution in the Web and other complex networks characterized by power laws. To this end, we model the PageRank as a solution of a stochastic equation where the Ris are distributed as R. This equation is inspired by the original definition of the PageRank. In particular, N models the number of incoming links to a page, and B stays for the user preference. Assuming that N or B are heavy tailed, we employ the theory of regular variation to obtain the asymptotic behavior of R under quite general assumptions on the involved random variables. Our theoretical predictions show good agreement with experimental data.

A large dam model is the object of study of this paper. The parameters Llower and Lupper define its lower and upper levels, L = Lupper - Llower is large, and if the current level of water is between these bounds, the dam is assumed to be in a normal state. Passage across one or other of the levels leads to damage. Let J1 and J2 denote the damage costs of crossing the lower and, respectively, the upper levels. It is assumed that the input stream of water is described by a Poisson process, while the output stream is state dependent. Let Lt denote the dam level at time t, and let p1 = limt→∞P{Lt = Llower} and p2 = limt→∞P{Lt > Lupper} exist. The long-run average cost, J = p1J1 + p2J2, is a performance measure. The aim of the paper is to choose the parameter controlling the output stream so as to minimize J.

The main results deal with conditions for the validity of the weighted convolution inequality ${{\sum }_{n\in \mathbb{Z}}}|{{b}_{n}}\,{{\sum }_{k\in \mathbb{Z}}}{{a}_{n-{{k}^{x}}k}}{{|}^{p}}\,\le \,{{C}^{p\,}}\,{{\sum }_{k\in \mathbb{Z}}}\,{{\left| {{x}_{k}} \right|}^{p}}$ when $p\,\ge \,1$.

Let ${{p}_{w}}(n)$ be the weighted partition function defined by the generating function $\Sigma _{n=0}^{\infty }{{p}_{w}}(n){{x}^{n}}=\prod{_{m=1}^{\infty }{{(1-{{x}^{m}})}^{-w(m)}}}$ , where $w\left( m \right)$ is a non-negative arithmetic function. Let ${{P}_{w}}(u)={{\Sigma }_{n\le u}}{{p}_{w}}(n)\,and\,{{N}_{w}}(u)={{\Sigma }_{n\le u}}w(n)$ be the summatory functions for ${{p}_{w}}(n)$ and $w\left( n \right)$, respectively. Generalizing results of G. A. Freiman and E. E. Kohlbecker, we show that, for a large class of functions $\Phi \left( u \right)$ and $\text{ }\!\!\lambda\!\!\text{ }\left( u \right)$, an estimate for ${{P}_{w}}\left( u \right)$ of the form log ${{P}_{w}}(u)=\Phi (u)\{1+Ou(1/\lambda (u))\}$$\left( u\to \infty \right)$ implies an estimate for ${{N}_{w}}(u)$ of the form ${{N}_{w}}(u)={{\Phi }^{*}}(u)\{1+O(1/\log \lambda (u))\}$$\left( u\to \infty \right)$ with a suitable function ${{\Phi }^{*}}(u)$ defined in terms of $\Phi \left( u \right)$. We apply this result and related results to obtain characterizations of the Riemann Hypothesis and the Generalized Riemann Hypothesis in terms of the asymptotic behavior of certain weighted partition functions.

Assume that for a measurable funcion f on (0, ∞) there exist a positive auxiliary function a(t) and some γ ∈ R such that . Then f is said to be of generalized regular variation. In order to control the asymptotic behaviour of certain estimators for distributions in extreme value theory we are led to study regular variation of second order, that is, we assume that exists non-trivially with a second auxiliary function a1(t). We study the possible limit functions in this limit relation (defining generalized regular variation of second order) and their domains of attraction. Furthermore we give the corresponding relation for the inverse function of a monotone f with the stated property. Finally, we present an Abel-Tauber theorem relating these functions and their Laplace transforms.

The summability fields of generalized Nörlund means (N,p*α,p), α ∈ Ν, are increasing with a and are contained in that of the corresponding power series method (P,p). Particular cases are the Cesàro- and Euler-means with corresponding power series methods of Abel and Borel. In this paper we generalize a convexity theorem, which is well-known for the Cesàro means and which was recently shown for the Euler means to a large class of generalized Nörlund means.

A new and very general and simple, yet powerful approach is introduced for obtaining new Tauberian theorems for a summability method V from known Tauberian conditions for V, where V is merely assumed to be linear and conservative. The technique yields the known theorems on the weakening of Tauberian conditions due to Meyer-König and Tietz and others and also improves many of them. Several new results are also obtained, even for classical methods of summability, including analogues of Tauber's second theorem for the Borel and logarithmic methods. The approach yields also new Tauberian conditions for the passage from summability by a method V to summability by a method V', as well as to more general methods of summability like absolute summability or summability in abstract spaces; the present paper however confines itself to ordinary summability.

In this paper we present a new approach to classical Karamata's results concerning the Hardy-Littlewood tauberian theorem.

Let t be a sequence in (0,1) that converges to 0, and define the Abel matrix At by ank = tn(1-tn)k. The matrix At determines a sequence-to-sequence variant of the classical Abel summability method. The purpose of this paper is to study these transformations as l-l summability methods: e.g., At maps l1 into l1 if and only if t is in l1. The Abel matrices are shown to be stronger l-l methods than the Euler-Knopp means and the Nӧrlund means. Indeed, if t is in l1 and Σ xk has bounded partial sums, then Atx is in l1. Also, the Abel matrix is shown to be translative in an l-l sense, and an l-l Tauberian theorem is proved for At.