Fourier Transformation of the Renormalization-Invariant Coupling

We discuss integral transformations of the QCD renormalization-invariant coupling (running coupling constant). Special attention is paid to the Fourier transformation, i.e., to the transition from the space–time to the energy–momentum representation. Our first conclusion is that the condition for the possibility of such a transition provides one more argument against the real existence of unphysical singularities observed in the perturbative QCD. The second conclusion relates to a way to translate some singular long-wave asymptotic behaviors to the infrared region of transferred momenta. Such a transition must be performed with the Tauberian theorem taken into account. This comment relates to the recent ALPHA collaboration results on the asymptotic behavior of the QCD effective coupling obtained by numerical lattice simulation.     

The Wiener-Ikehara theorem by complex analysis

The Tauberian theorem of Wiener and Ikehara provides the most direct way to the prime number theorem. Here it is shown how Newman's contour integration method can be adapted to establish the Wiener-Ikehara theorem. A simple special case suffices for the PNT. But what about the twin-prime problem?

     

A family of new trigonometric means

The object of the present paper is to introduce a new family of trigonometric means which are regular. Many interesting inclusion relations with well-known nontrigonometric means and lastly a Tauberian theorem have been established.     

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.     

Abelian and Tauberian theorems for the Laplace transform of functions in several variables

Using two kinds of multivariate regular variation we prove several Abel-Tauber theorems for the Laplace transform of functions in several variables. We generalize some power series results of Alpar and apply our results in multivariate renewal theory.     

Tauberian Theorems for Sequences Satisfying Certain Convolution Relations

The purpose of this paper is to prove qualitative and optimal quantitative Tauberian theorems for the sequence λ defined by the relation g′ = λ∗︁g ,where ∗︁ denotes the Cauchy convolution, g is a given sequence with g(0) = 0 satisfying suitable growth conditions, and g′ is defined by g′(n) = (n + 1)g(n + 1) for n ∈ ℕ = {0, 1, 2, …}. The problem originates in the analytic theory of the distribution of prime elements in additive arithmetic semigroups developed by Knopfmacher , and in recent work of Indlekofer , Manstavičius and Warlimont , of Zhang , and of Warlimont . The new results are based on the approximation of g by recurrent sequences and do not assume λ ≥ 0; the method of proof consists in the transition to generating power series and utilizes a weighted version of a Wiener type inversion theorem due to Lucht and Reifenrath .     

The leading term of the spectral asymptotics for the Kohn-Laplace operator in a bounded domain

We prove the Weyl asymptotic formula for the number of eigenvalues of the Kohn-Laplace operator on a Heisenberg group and write out the leading term of asymptotics. The method of study is based on estimates of the Green function for the Dirichlet problem for the corresponding parabolic operator and makes use of the classical Hardy-Littlewood Tauberian theorem.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 493–505, October, 1998.This research was supported by the Russian Foundation for Basic Research under grants No. 96-01-00443 and No. 96-01-00503.     

Exact tail asymptotics for the Israeli queue with retrials and non-persistent customers

In this paper, we consider the Israeli queue which consists of a main queue with at most N groups and an infinite capacity retrial orbit. The retrial customers may become non-persistent before receiving service. This model was considered before and the decay rate function of the stationary distribution was obtained. To strengthen the result, we characterize the exact tail asymptotics by calculating the coefficient before the decay rate function.     

A one-sided Tauberian theorem for the Borel summability method

We establish a quantitative version of Vijayaraghavan's classical result and use it to give a short proof of the known theorem that a real sequence (s_n) which is summable by the Borel method, and which satisfies the one-sided Tauberian condition that is bounded below must be convergent.     

Tauberian Theorem for the Stieltjes Transform

Under weak constraints on the positive functions to be compared, we derive their asymptotic equivalence at infinity as a consequence of the asymptotic equivalence of their Stieltjes transforms at infinity.     

Asymptotic behavior of solutions to abstract functional differential equations

In this paper, a new approach is provided to study the asymptotic behavior of functions. A Tauberian theorem is improved and applied to describe the asymptotic behavior of abstract functional differential equations of the form

     

Asymptotics for logical limit laws: When the growth of the components is in an RT class

Compton's method of proving monadic second-order limit laws is based on analyzing the generating function of a class of finite structures. For applications of his deeper results we previously relied on asymptotics obtained using Cauchy's integral formula. In this paper we develop elementary techniques, based on a Tauberian theorem of Schur, that significantly extend the classes of structures for which we know that Compton's theory can be applied.

     

A unified framework for harmonic analysis of functions on directed graphs and changing data

We present a general framework for studying harmonic analysis of functions in the settings of various emerging problems in the theory of diffusion geometry. The starting point of the now classical diffusion geometry approach is the construction of a kernel whose discretization leads to an undirected graph structure on an unstructured data set. We study the question of constructing such kernels for directed graph structures, and argue that our construction is essentially the only way to do so using discretizations of kernels. We then use our previous theory to develop harmonic analysis based on the singular value decomposition of the resulting non-self-adjoint operators associated with the directed graph. Next, we consider the question of how functions defined on one space evolve to another space in the paradigm of changing data sets recently introduced by Coifman and Hirn. While the approach of Coifman and Hirn requires that the points on one space should be in a known one-to-one correspondence with the points on the other, our approach allows the identification of only a subset of landmark points. We introduce a new definition of distance between points on two spaces, construct localized kernels based on the two spaces and certain interaction parameters, and study the evolution of smoothness of a function on one space to its lifting to the other space via the landmarks. We develop novel mathematical tools that enable us to study these seemingly different problems in a unified manner.     

Ratio limit theorems and Tauberian theorems for vector-valued functions and sequences

We prove ratio limit theorems for (C,γ)-means (γ?0) and Abel means of functions and sequences in Banach spaces, and ratio Tauberian theorems for (C,γ)-means (γ?1) and Abel means of functions and sequences in Banach lattices.     

Tauberian theorem for m‐spherical transforms on the Heisenberg group

In this paper we prove a Tauberian type theorem for the space L ( H _n ). This theorem gives sufficient conditions for a L ( H _n ) submodule J ? L ( H _n ) to make up all of L ( H _n ). As a consequence of this theorem, we are able to improve previous results on the Pompeiu problem with moments on the Heisenberg group for the space L^∞( H _n ). In connection with the Pompeiu problem, given the vanishing of integrals ∫ z ^m L _g f ( z , 0) dσ ( z ) = 0 for all g ∈ H _n and i = 1, 2 for appropriate radii r₁ and r₂, we now have the (improved) conclusion f ≡ 0, where = · · · and form the standard basis for T^(0,1)( H _n ). (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Statistical extensions of some classical Tauberian theorems

Hardy's well-known Tauberian theorem for Cesàro means says that if the sequence satisfies and , then . In this paper it is shown that the hypothesis can be replaced by the weaker assumption of the statistical limit: st-lim , i.e., for every , . Similarly, the ``one-sided' Tauberian theorem of Landau and Schmidt's Tauberian theorem for the Abel method are extended by replacing and with st-lim and st-lim , respectively. The Hardy-Littlewood Tauberian theorem for Borel summability is also extended by replacing , where is a continuous parameter, with , and further replacing it by -st-lim , where is the Borel matrix method.

     

Tauberian Theorems for Sequences linked by a Convolution

Given sequences g, λ: IN₀ → C, g (0) = 1, linked by the convolution λ g = g′¹ (g′¹(n)= (n + 1) g (n + 1)) we study what can be inferred about λ (n → ∞) from some concrete information about the behaviour of g (n → ∞).     

On the summatory function of a class of multiplicative functions

We consider an asymptotic formula for the mean value of one class of multiplicative functions, the error term in this formula being related to an estimate of its short interval analogue. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 1, pp. 111–122, January–March, 2006.     

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.