Tauberian Results Research Articles

The shearlet transform and asymptotic behavior of Lizorkin distributions

ABSTRACT In this paper, we establish Abelian and Tauberian results that characterize the quasiasymptotic behavior of Lizorkin distributions via the asymptotic behavior of their shearlet transform.

Abelian and Tauberian results for the fractional Fourier cosine (sine) transform

<abstract><p>In this paper, we presented Tauberian type results that intricately link the quasi-asymptotic behavior of both even and odd distributions to the corresponding asymptotic properties of their fractional Fourier cosine and sine transforms. We also obtained a structural theorem of Abelian type for the quasi-asymptotic boundedness of even (resp. odd) distributions with respect to their fractional Fourier cosine transform (FrFCT) (resp. fractional Fourier sine transform (FrFST)). In both cases, we quantified the scaling asymptotic properties of distributions by asymptotic comparisons with Karamata regularly varying functions.</p></abstract>

Abelian and Tauberian results for the fractional Fourier and short-time Fourier transforms of distributions

ABSTRACT We introduce the fractional short-time Fourier transform in and provide generalized asymptotics for the fractional Fourier and the short-time Fourier transforms within . Abelian- and Tauberian-type results are given.

Estimates for the Weyl coefficient of a two-dimensional canonical system

For a two-dimensional canonical system $y'(t)=zJH(t)y(t)$ on some interval $(a,b)$ whose Hamiltonian $H$ is a.e. positive semi-definite and which is regular at $a$ and in the limit point case at $b$, denote by $q_H$ its Weyl coefficient. De~Branges' inverse spectral theorem states that the assignment $H\mapsto q_H$ is a bijection between Hamiltonians (suitably normalised) and Nevanlinna functions. We give upper and lower bounds for $|q_H(z)|$ and $\Im q_H(z)$ when $z$ tends to $i\infty$ non-tangentially. These bounds depend on the Hamiltonian $H$ near the left endpoint $a$ and determine $|q_H(z)|$ up to universal multiplicative constants. We obtain that the growth of $|q_H(z)|$ is independent of the off-diagonal entries of $H$ and depends monotonically on the diagonal entries in a natural way. The imaginary part is, in general, not fully determined by our bounds (in forthcoming work we shall prove that for most Hamiltonians also $\Im q_H(z)$ is fully determined). We translate the asymptotic behaviour of $q_H$ to the behaviour of the spectral measure $\mu_H$ of $H$ by means of Abelian--Tauberian results and obtain conditions for membership of growth classes defined by weighted integrability condition (Kac classes) or by boundedness of tails at $\pm\infty$ w.r.t. a weight function. Moreover, we apply our results to Krein strings and Sturm--Liouville equations.

Non‐classical Tauberian and Abelian type criteria for the moment problem

AbstractThe aim of this paper is to provide some new criteria for the determinacy problem of the Stieltjes moment problem. We first give a Tauberian type criterion for moment indeterminacy that is expressed purely in terms of the asymptotic behavior of the moment sequence (and its extension to imaginary lines). Under an additional assumption this provides a converse to the classical Carleman's criterion, thus yielding an equivalent condition for moment determinacy. We also provide a criterion for moment determinacy that only involves the large asymptotic behavior of the distribution (or of the density if it exists), which can be thought of as an Abelian counterpart to the previous Tauberian type result. This latter criterion generalizes Hardy's condition for determinacy, and under some further assumptions yields a converse to the Pedersen's refinement of the celebrated Krein's theorem. The proofs utilize non‐classical Tauberian results for moment sequences that are analogues to the ones developed in [8] and [3] for the bi‐lateral Laplace transforms in the context of asymptotically parabolic functions. We illustrate our results by studying the time‐dependent moment problem for the law of log‐Lévy processes viewed as a generalization of the log‐normal distribution. Along the way, we derive the large asymptotic behavior of the density of spectrally‐negative Lévy processes having a Gaussian component, which may be of independent interest.

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.

Tauberian class estimates for vector-valued distributions

We study Tauberian properties of regularizing transforms of vector-valued tempered distributions. The transforms have the form , where the kernel is a test function and . We investigate conditions which ensure that a distribution that a priori takes values in a locally convex space actually takes values in a narrower Banach space. Our goal is to characterize spaces of Banach-space-valued tempered distributions in terms of so-called class estimates for the transform . Our results generalize and improve earlier Tauberian theorems due to Drozhzhinov and Zav’yalov. Special attention is paid to finding the optimal class of kernels for which these Tauberian results hold. Bibliography: 24 titles.

On exact Hausdorff measure functions of operator semistable Lévy processes

ABSTRACTLet X = {X(t)}t ⩾ 0 be an operator semistable Lévy process on with exponent E, where E is an invertible linear operator on . In this article, we determine exact Hausdorff measure functions for the range of X over the time interval [0, 1] under certain assumptions on the principal spectral component of E. As a byproduct, we also present Tauberian results for semistable subordinators and sharp bounds for the asymptotic behavior of the expected sojourn times of X.

Abelian and Tauberian results for the one-dimensional modified Stockwell transforms

The aim of this paper is to study the asymptotic behavior of one- dimensional modified Stockwell transform of a tempered distribution signal through the quasiasymptotic behavior at origin or infinity of the signal itself. More precisely, we give some Abelian results which mean that we derive the asymptotic properties of the S-transform of a tempered signal from the quasiasymptotic properties of the signal itself and we do also the opposite. So, we also give some Tauberian results which describe some quasiasymptotic properties of the tempered signal by means of the asymptotic properties of its Stockwell transform.

On the logarithmic summability method for sequences of fuzzy numbers

We define logarithmic summability method for sequences of fuzzy numbers and prove theorems dealing with the convergence behavior of logarithmic summable sequences of fuzzy numbers. The study also reveals slowly decreasing and Landau one-sided type Tauberian results analogous to those given by Moricz (Stud Math 219:109–121, 2013).

On the Borel summability method of sequences of fuzzy numbers

In the current paper, we define Borel summability of sequences of fuzzy numbers and investigate Tauberian conditions under which Borel summable sequences of fuzzy numbers are convergent. Also, we give analogous Tauberian results for Borel summability method of series of fuzzy numbers.

Abel-type Results for Controlled Piecewise Deterministic Markov Processes

In this short paper we prove that, in the framework of continuous control problems for piecewise deterministic Markov processes, the existence of a uniform limit for discounted value functions as the discount factor vanishes implies (without any further assumption) the uniform convergence of the value functions with long run average cost as the time horizon increases to infinity. The two limit values coincide. We also provide a converse Tauberian result for a particular class of systems with Poisson-triggered jump mechanism. We exhibit a very simple example in which the dynamics are not dissipative, nevertheless discounted values converge uniformly to a non-constant limit function.

Abel summability of sequences of fuzzy numbers

In the present study, we have introduced the concept of Abel summability for sequences and series of fuzzy numbers. Also, some tauberian results in classical analysis have been generalized to fuzzy analysis.

Large deviations for the boundary local time of doubly reflected Brownian motion

We compute a closed-form expression for the moment generating function fˆ(x;λ,α)=1λEx(eαLτ), where Lt is the local time at zero for standard Brownian motion with reflecting barriers at 0 and b, and τ∼Exp(λ) is independent of W. By analyzing how and where fˆ(x;⋅,α) blows up in λ, a large-time large deviation principle (LDP) for Lt/t is established using a Tauberian result and the Gärtner–Ellis Theorem.

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 apply these Tauberian results to deduce a number of Tauberian theorems for power series where Ces\`{a}ro 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.

Generalizations of Fatou's and Marcinkiewicz's results

Two new criteria for almost everywhere, uniform, and L 1 -convergence of Fourier se- ries are established. The first one is a Tauberian result for (C,1) summability, which generalizes a result of Fatou. The second one is a norm analogue of Dini's test, which generalizes a re- sult of Marcinkiewicz. It is also pointed out that our second result is not comparable with the Dini-Lipschitz test.

A quick distributional way to the prime number theorem

We use distribution theory (generalized functions) to show the prime number theorem. No tauberian results are employed.

Tauberian conditions for w-almost convergent double sequences

In this paper, we introduce the concept of w-almost convergent sequences. Such a definition is a weak form of almost convergent sequences given by G. G. Lorentz in [Acta Math. 80(1948),167-190]. We give a detailed study on w-almost convergent double sequences and prove that w-almost convergence and almost convergence are equivalent under the boundedness of the given sequence. The Tauberian results for w-almost convergence are established. Our Tauberian results generalize a result of Lorentz and Tauber’s second theorem. Moreover, we prove that w-almost convergence and norm convergence are equivalent for the sequence of the rectangular partial sums of the Fourier series of f ∈ Lp(T2), where 1 < p < ∞.

Tauberian conditions for almost convergence

In [Acta Math. 80(1948), 167–190], G. G. Lorentz characterized almost convergent sequences in \(\mathbb{R}\) (or in \(\mathbb{C}\)) in terms of the concept of uniform convergence of the de la Vallee-Poussin means. In this paper, we present Tauberian results which relate almost convergence to norm convergence or to the (C, 1) convergence. Our results generalize Kronecker lemma. As a consequence, we prove that almost convergence and norm convergence are equivalent for the sequence of the partial sums of the Fourier series of \(f \in L^p(T)\) (or \(f \in C(T)\)), where \(1\leq p \leq \infty\). We also show that our results can be used to derive Fatou’s theorem.

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 Cesaro-mean-bounded semigroups (discrete and continuous) of operators and for semigroups of positive operators.