Tauberian Theorems Research Articles

Tauberian theorems for smooth functions that never vanish

It is well known that if a function has slow growth in the ordinary sense, then it has slow growth in the Cesàro sense. The converse holds if special extra conditions are assumed; results of this form are called Tauberian Theorems. In this paper, we use techniques from generalized functions to investigate this phenomenon, with particular emphasis on the space of multipliers of Schwartz functions. We then apply our results to study series of delta functions and the division problem for tempered distributions.

Polynomial approximations in a generalized Nyman–Beurling criterion

The Nyman-Beurling criterion, equivalent to the Riemann hypothesis, is an approximation problem in the space of square integrable functions on $(0,\infty)$, involving dilations of the fractional part function by factors $\theta_k\in(0,1)$, $k\ge1$. Randomizing the $\theta_k$ generates new structures and criteria. One of them is a sufficient condition that splits into (i) showing that the indicator function can be approximated by convolution with the fractional part, (ii) a control on the coefficients of the approximation. This self-contained paper aims at identifying functions for which (i) holds unconditionally, by means of polynomial approximations. This yields in passing a short probabilistic proof of a known consequence of Wiener's Tauberian theorem. In order to tackle (ii) in the future, we give some expressions of the scalar products. New and remarkable structures arise for the Gram matrix, in particular moment matrices for a suitable weight that may be the squared $\Xi$-function for instance.

An Operator Theoretic Approach to the Prime Number Theorem

In this short note, we establish an operator theoretic version of the Wiener-Ikehara tauberian theorem, and point out how this leads to a new proof of the Prime number theorem that should be accessible to anyone with a basic knowledge of operator theory.

Abelian theorems involving the fractional wavelet transforms

In this paper, the initial-value and the final-value Abelian theorems are presented for the continuous fractional wavelet transform of functions and distributions. An application of these Abelian theorems to the continuous fractional wavelet transforms is also investigated by using the Mexican hat wavelet function.

Operators with complex Gaussian kernels: Asymptotic behaviours

In this paper we derive Abelian theorems for the operators with complex Gaussian kernels. Specifically, we establish some results in which known the behaviour of the function and its domain variable approaches to ?? or +? is used to infer the asymptotic behaviour of the transform as its domain variable approaches to +? or ??. For this purpose we use a formula concerning the computation of potential functions by means of these operators with complex Gaussian kernels. This formula allows us to analyse the asymptotic behaviour of these operators in both cases: when the variable approaches to +? or ??. Our results include systematically the noncentered and centered cases of these operators. Here we analyse the Gauss-Weierstrass semigroup on R as a particular case. We also point out Abelian theorems for other kinds of operators which have been studied in several papers.

Tauberian theorems for the Cesàro summability method of regularly generated double integrals

For a continuous function 1 over R2+ := [1,?) ? [1,?), we denote its integral over [1, x] ? [1, y] by h(x, y) = R x 1 R y 1 1(u, v)dudv and its (C, 1, 1) mean, the average of h(x, y) over [1, x] ? [1, y], by t(h(x, y)) = (xy)?1 R x 1 R y 1 h(u, v)dudv. Analogously, the other means (C, 1, 0) and (C, 0, 1) can be defined. In this paper, we introduce the concept of regularly generated double integrals in senses (1, 1), (1, 0) and (0, 1) and obtain Tauberian conditions in terms of the regularly generated double integrals in senses (1,1), (1,0) and (0,1) under which convergence of h(x, y) follows from that of t(h(x,y)).

Deferred Cesàro means of fuzzy number-valued sequences with applications to Tauberian theorems

In this paper, the deferred Ces?ro means of fuzzy number-valued sequences are studied and their summability by the deferred Ces?ro method with respect to the supremum metric is introduced. Also, Tauberian conditions to retrieve the convergence of a fuzzy number-valued sequence from its deferred Ces?ro summability are investigated.

Abelian theorems for the index 2F1-transform over distributions of compact support and generalized functions

Abelian theorems for the index 2F1-transform over distributions of compact support and generalized functions

Quadratic fields admitting elliptic curves with rational j-invariant and good reduction everywhere

Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by x over which there exist elliptic curves with good reduction everywhere and rational j-invariant is ≫xlog−12⁡(x). In this paper, we assume the abc-conjecture to show the sharp asymptotic ∼cxlog−12⁡(x) for this number, obtaining formulae for c in both the real and imaginary cases. Our method has three ingredients:(1)We make progress towards a conjecture of Granville: Given a fixed elliptic curve E/Q with short Weierstrass equation y2=f(x) for reducible f∈Z[x], we show that the number of integers d, |d|≤D, for which the quadratic twist dy2=f(x) has an integral non-2-torsion point is at most D23+o(1), assuming the abc-conjecture.(2)We apply the Selberg–Delange method to obtain a Tauberian theorem which allows us to count integers satisfying certain congruences while also being divisible only by certain primes.(3)We show that for a polynomially sparse subset of the natural numbers, the number of pairs of elements with least common multiple at most x is O(x1−ϵ) for some ϵ>0. We also exhibit a matching lower bound. If instead of the abc-conjecture we assume a particular tail bound, we can prove all the aforementioned results and that the coefficient c above is greater in the real quadratic case than in the imaginary quadratic case, in agreement with an experimentally observed bias.

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 ordinary convergence

We show that a real sequence x is convergent if and only if there exist a regular matrix A and an Fσδ-ideal I on N such that the set of subsequences y of x for which Ay is I-convergent is of the second Baire category. This includes the cases where I is the ideal of asymptotic density zero sets, the ideal of Banach density zero sets, and the ideal of finite sets. The latter recovers an old result given by Keogh and Petersen in (1958) [26]. Our proofs are of a different nature and rely on recent results in the context of I-Baire classes and filter games.As application, we obtain a stronger version of the classical Steinhaus' theorem: for each regular matrix A, there exists a {0,1}-valued sequence x such that Ax is not statistically convergent.

Tauberian Korevaar

We focus on the Tauberian work for which Jaap Korevaar is best known, together with its connections with probability theory. We begin with a brief sketch of the field up to Beurling’s work. We follow with three sections on Beurling aspects: Beurling slow variation; the Beurling Tauberian theorem for which it was developed; Riesz means and Beurling moving averages. We then give three applications from probability theory: extremes, laws of large numbers, and large deviations. We then turn briefly to other areas of Korevaar’s work. We close with a personal postscript (whence our title).

A noncommutative Tauberian theorem and Weyl asymptotics in noncommutative geometry

A noncommutative Tauberian theorem and Weyl asymptotics in noncommutative geometry

Abelian theorems for Laplace, Mellin and Stieltjes transforms over distributions of compact support and generalized functions

Abelian theorems for Laplace, Mellin and Stieltjes transforms over distributions of compact support and generalized functions

Determining damping terms in fractional wave equations

This paper deals with the inverse problem of recovering an arbitrary number of fractional damping terms in a wave equation. We develop several approaches on uniqueness and reconstruction, some of them relying on Tauberian theorems that provide relations between the asymptotic behaviour of solutions in time and Laplace domains. The possibility of additionally recovering space-dependent coefficients or initial data is discussed. The resulting methods for reconstructing coefficients and fractional orders in these terms are tested numerically. In addition, we provide an analysis of the forward problem consisting of a multiterm fractional wave equation.

New Tauberian theorems for statistical Cesàro summability of a function of three variables over a locally convex space

In this paper, we prove two new Tauberian theorems via statistical Cesàro summability mean of a continuous function of three variables by using oscillating behavior and De la Vallée Poussin means of a triple integral over a locally convex space. Moreover, some remarks and corollaries are provided here to support our results.

A Tauberian approach to an analog of Weyl’s law for the Kohn Laplacian on compact Heisenberg manifolds

Let \(M= \Gamma \setminus \mathbb {H}_d\) be a compact quotient of the d-dimensional Heisenberg group \(\mathbb {H}_d\) by a lattice subgroup \(\Gamma \). We show that the eigenvalue counting function \(N^\alpha \left( \lambda \right) \) for any fixed element of a family of second order differential operators \(\left\{ \mathcal {L}_\alpha \right\} \) on M has asymptotic behavior \(N^\alpha \left( \lambda \right) \sim C_{d,\alpha } {\text {vol}}\left( M\right) \lambda ^{d + 1}\), where \(C_{d,\alpha }\) is a constant that only depends on the dimension d and the parameter \(\alpha \). As a consequence, we obtain an analog of Weyl’s law (both on functions and forms) for the Kohn Laplacian on M. Our main tools are Folland’s description of the spectrum of \({\mathcal {L}}_{\alpha }\) and Karamata’s Tauberian theorem.

On extremes of PSI-processes and gaussian limits of their normalized independent identical distributed sums

We define PSI-process — Poisson Stochastic Index process, as a continuous time random process which is obtained by a manner of a randomization for the discrete time of a random sequence. We consider the case when a double stochastic Poisson process generates this randomization, i. e. such Poisson process has a random intensity. Under condition of existence of the second moment the stationary PSI-processes possess a covariance which coincides with the Laplace transform of the random intensity. In our paper we derive distributions of extremes for a one PSI-process, and these extremes are expressed in terms of Laplace transform of the random intensity. The second task that we solve is a convergence of the maximum of Gaussian limit for normalized sums of i. i. d. stationary PSI-processes. We obtain necessary and sufficient conditions for the intensity under which, after proper centering and normalization, this Gaussian limit converges in distribution to the double Exponential Law. For solution this task we essentially base on the monograph: M.R.Leadbetter, Georg Lindgren, Holder Rootzen (1986) “Extremes and Relative Properties of Random Sequences and Processes”, end essentially use the Tauberian theorem in W. Feller form.

Tauberian theorems via the generalized Nörlund mean for sequences in 2-normed spaces

Tauberian theorems via the generalized Nörlund mean for sequences in 2-normed spaces

Tauberian theorems for weighted means of double sequences in intuitionistic fuzzy normed spaces

We define weighted mean summability method of double sequences in intuitionistic fuzzy normed spaces(IFNS), and obtain necessary and sufficient Tauberian conditions under which convergence of double sequences in IFNS follows from their weighted mean summability. This study reveals also Tauberian results for some known summation methods in the special cases.