Tauberian theorems for weighted means of double sequences

We discuss the relations between weighted mean methods and ordinary convergence for double sequences. In particular, we study Tauberian theorems also for methods not being products of the related one-dimensional summability methods. For the special case of theC _1,1-method, the results contain a classical Tauberian theorem by Knopp [9] as special case and generalize theorems given by Móricz [16] thereby showing that one of his Tauberian conditions can be weakened.     

Summability in topological spaces

The main purpose of the paper is to introduce the notion of summability in abstract Hausdorff topological spaces. We give a characterization of such summability methods when the space allows a countable base. We also provide several Tauberian theorems in topological structures. Some open problems are discussed.     

Some Special Tauberian Theorems for Fourier Type Integrals and Eigenfunction Expansions of Sturm-Liouville Equations on the Whole Line

We offer a new proof of a special Tauberian theorem for Fourier type integrals. This Tauberian theorem was already considered by us in the papers [1] and [2]. The idea of our initial proof was simple, but the details were complicated because we used Bochner's definition of generalized Fourier transform for functions of polynomial growth. In the present paper we work with L. Schwartz's generalization. This leads to significant simplification. The paper consists of six sections. In Section 1 we establish an integral representation of functions of polynomial growth (subjected to some Tauberian conditions), in Section 2 we prove our main Tauberian theorems (Theorems 2.1 and 2.2.), using the integral representation of Section 1, in Section 3 we study the asymptotic behavior of M. Riesz's means of functions of polynomial growth, in Sections 4 and 5 we apply our Tauberian theorems to the problem of equiconvergence of eigenfunction expansions of Sturm-Liouville equations and expansion in ordinary Fourier integrals, and in Section 6 we compare our general equiconvergence theorems of Sections 4 and 5 with the well known theorems on eigenfunction expansions in classical orthogonal polynomials. In some sense this paper is a re-made survey of our results obtained during the period 1953-58. Another proof of our Tauberian theorem and some generalization can be found in the papers [3] and [4].     

Distributional versions of littlewood’s Tauberian theorem

We provide several general versions of Littlewood’s Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We employ two types of Tauberian hypotheses; the first kind involves distributional boundedness, while the second type imposes a one-sided assumption on the Cesàro behavior of the distribution. We apply these Tauberian results to deduce a number of Tauberian theorems for power series and Stieltjes integrals where Cesàro summability follows from Abel summability. We also use our general results to give a new simple proof of the classical Littlewood one-sided Tauberian theorem for power series.     

On Tauberian theorems for (A)(C, α) summability method

In this paper we introduce some Tauberian conditions for the (A)(C, α) summability method. These results extend and generalize some of the classical Tauberian theorems for the Abel summability method.     

Tauberian conditions for Cesàro summability of integrals

In this paper we generalize some classical type Tauberian theorems given for Cesàro summability of integrals.     

Ratio Tauberian Theorems for Positive Functions and Sequences in Banach Lattices

We prove two ratio Tauberian theorems and deduce two generalized Tauberian theorems for functions and sequences with values in positive cones of Banach lattices. Two counter-examples are given to show that the hypotheses in the ratio Tauberian theorems are essential.     

Tauberian Type Conditions Connecting Cesàro and Riesz Summability Methods of Different Orders

In the paper conditions are studied which ensure the equivalence of two given summability methods on the class of numerical series satisfying these conditions. One of these methods is the regular Cesàro method (C,w), which replaces the role of the convergence (C,0) the latter occurring in the classical Tauberian theorems. The other summability method considered is either the summability method (Rd,) by discrete Riesz means or the Cesàro method with (Rd,) with > w. New Tauberian type conditions are obtained under which the corresponding Tauberian theorems are proved.     

Tauberian theorems with remainder for (H, p, α, β)-and (C, p, α, β)-methods of summation of functions of two variables

We consider a general method of obtaining Tauberian theorems with remainder for Hölder- and Cesarotype methods of summation.     

Asymptotically exhaustive functions in Vitali spaces

We introduce asymptotically exhaustive functions defined on Vitali Spaces with values in a Hausdorff commutative topological group and we prove for them some classical convergence theorems. This article was supported by MURST, project “Analisi Reale”     

Comparison of certain summability methods by speeds of convergence

We deal with families of summability methods which depend on a continuous parameter and where two different methods are connected either by a Cesàro-type or Euler–Knopp-type method. Extending and applying some results of [14], we are able to compare speeds of convergence in families of generalized Nörlund methods and to give certain Tauberian remainder theorems. Particular cases are the families of Cesàro, generalized Cesàro and Euler–Knopp methods.     

Statistic D-Property of Voronoi Summation Methods of Class W Q 2

We propose a general method for obtaining Tauberian theorems with remainder for one class of Voronoi summation methods for double sequences of elements of a locally convex, linear topological space. This method is a generalization of the Davydov method of C-points.     

Convergence theorems and Tauberian theorems for functions and sequences in Banach spaces and Banach lattices

We prove generalized convergence theorems and Tauberian theorems for vector-valued functions and sequences of growth order γ − 1 with γ > 0 and for positive functions and sequences in Banach lattices. Then the general results are applied to obtain some interesting particular Tauberian results for various examples of operator semigroups. Among them are mean ergodic theorems for Cesàro-mean-bounded semigroups (discrete and continuous) of operators and for semigroups of positive operators. Research supported in part by the National Science Council of Taiwan. Current address: 19-18, Higashi-hongo 2-chome, Midori-ku, 226-0002 Japan.     

Limit shapes of young diagrams. Two elementary approaches

Techniques are presented for obtaining the limit shapes of Young diagrams with respect to multiplicative measures, which arise in statistical mechanics. The approach employs neither complex analysis nor Tauberian theorems. Also, the limit shape is found for bounded and unbounded partitions with respect to the uniform measure, without using even generating functions. Bibliography: 6 titles.     

Harmonic and Logarithmic Summability of Orthogonal Series are Equivalent up to a Set of Measure Zero

We prove Tauberian theorems from J_p-summability methods of powerseries type to ordinary convergence, respectively M_p-summabilitymethods of weighted means. Particular cases are the Abel andCesàro, as well as logarithmic and harmonic summability.Besides numerical series, we also consider orthogonal serieswith coefficients from L₂. In the latter case, it turns outthat one of our Tauberian conditions is satisfied almost everywhereon the underlying measure space, thereby proving the claim statedin the title.     

A finitization of Littlewood's Tauberian theorem and an application in Tauberian remainder theory

In this paper we study Littlewood's Tauberian theorem from a proof theoretic perspective. We first use the Dialectica interpretation to produce an equivalent, finitary formulation of the theorem, and then carry out an analysis of Wielandt's proof to extract concrete witnessing terms. We argue that our finitization can be viewed as a generalized Tauberian remainder theorem, and we instantiate it to produce two concrete remainder theorems as a corollary, in terms of rates of convergence and rates metastability, respectively. We rederive the standard remainder estimate for Littlewood's theorem as a special case of the former.     

Tauberian theorems for the (J,p) summability method

In this paper, we prove Tauberian theorems of slowly oscillating type for the $(J,p)$ summability method by using a result due to Mikhalin [G.A. Mikhalin, Theorem of Tauberian type for (J, pn) summation methods, Ukrain. Mat. Zh. 29 (1977), 763–770. English translation: Ukrain. Math. J. 29 (1977) 564–569].     

Applications of Tauberian theorems in some problems in mathematical physics

We give some multidimensional Tauberian theorems for generalized functions and show examples of their application in mathematical physics. In particular, we consider the problems of stabilizing the solutions of the Cauchy problem for the heat kernel equation, multicomponent gas diffusion, and the asymptotic Cauchy problem for a free Schrödinger equation in the norms of different Banach spaces among others.     

One-sided Tauberian conditions and double sequences

We give a theorem of Vijayaraghavan type for summability methods for double sequences, which allows a conclusion from boundedness in a mean and a one-sided Tauberian condition to the boundedness of the sequence itself. We apply the result to certain power series methods for double sequences improving a recent Tauberian result by S. Baron and the author [4]. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Complement to the Valiron-Titchmarsh Theorem for Subharmonic Functions

The Valiron-Titchmarsh theorem on asymptotic behavior of entire functions with negative zeros has been recently generalized onto subharmonic functions with the Riesz measure on a half-line in Rn, n ≥ 3. Here we extend the Drasin complement to the Valiron-Titchmarsh theorem and show that if u is a subharmonic function of this class and of order 0 ρ 1, then the existence of the limit limr →∞ logu(r)/N(r),where N(r) is the integrated counting function of the masses of u, implies the regular asymptotic behavior for both u and its associated measure.