Memory estimation of inverse operators

We use methods of harmonic analysis and group representation theory to estimate the memory decay of the inverse operators in Banach spaces. The memory of the operators is defined using the notion of the Beurling spectrum. We obtain a general continuous non-commutative version of the celebrated Wiener's Tauberian Lemma with estimates of the “Fourier coefficients” of inverse operators. In particular, we generalize various estimates of the elements of the inverse matrices. The results are illustrated with a variety of examples including integral and integro-differential operators.     

Density of irregular wavelet frames

We show that if an irregular multi-generated wavelet system forms a frame, then both the time parameters and the logarithms of scale parameters have finite upper Beurling densities, or equivalently, both are relatively uniformly discrete. Moreover, if generating functions are admissible, then the logarithms of scale parameters possess a positive lower Beurling density. However, the lower Beurling density of the time parameters may be zero. Additionally, we prove that there are no frames generated by dilations of a finite number of admissible functions.

     

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.     

On a tauberian theorem of Hardy and Littlewood

In this paper, we give a simple alternative proof of a Tauberian theorem of Hardy and Littlewood (Theorem E stated below, [3]).     

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.     

First and higher order uniform dual ergodic theorems for dynamical systems with infinite measure

We generalize the proof of Karamata’s Theorem by the method of approximation by polynomials to the operator case. As a consequence, we offer a simple proof of uniform dual ergodicity for a very large class of dynamical systems with infinite measure, and we obtain bounds on the convergence rate. In many cases of interest, including the Pomeau-Manneville family of intermittency maps, the estimates obtained through real Tauberian remainder theory are very weak. Building on the techniques of complex Tauberian remainder theory, we develop a method that provides second (and higher) order asymptotics. In the process, we derive a higher order Tauberian theorem for scalar power series which, to our knowledge, has not previously been covered.     

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.

     

Weak amenability and 2-weak amenability of Beurling algebras

Let be a Beurling algebra on a locally compact abelian group G. We look for general conditions on the weight which allows the vanishing of continuous derivations of . This leads us to introducing vector-valued Beurling algebras and considering the translation of operators on them. This is then used to connect the augmentation ideal to the behavior of the derivation space. We apply these results to give examples of various classes of Beurling algebras which are weakly amenable, 2-weakly amenable or fail to be even 2-weakly amenable.     

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.     

Tail probability of random variable and Laplace transform

We investigate the exponential decay of the tail probability P(X?>?x) of a continuous type random variable X. Let ?(s) be the Laplace–Stieltjes transform of the probability distribution function F(x)?=?P(X?≤?x) of X, and σ₀ be the abscissa of convergence of ?(s). We will prove that if ?∞?0?s) on the axis of convergence are only a finite number of poles, then the tail probability decays exponentially. For the proof of our theorem, Ikehara's Tauberian theorem will be extended and applied.     

Beurling Dimension of Gabor Pseudoframes for Affine Subspaces

Pseudoframes for subspaces have been recently introduced by Li and Ogawa as a tool to analyze lower dimensional data with arbitrary flexibility of both the analyzing and the dual sequence. In this paper we study Gabor pseudoframes for affine subspaces by focusing on geometrical properties of their associated sets of parameters. We first introduce a new notion of Beurling dimension for discrete subsets of ℝ ^d by employing a certain generalized Beurling density. We present several properties of Beurling dimension including a comparison with other notions of dimension showing, for instance, that our notion includes the mass dimension as a special case. Then we prove that Gabor pseudoframes for affine subspaces satisfy a certain Homogeneous Approximation Property, which implies invariance under time–frequency shifts of an approximation by elements from the pseudoframe. The main result of this paper is a classification of Gabor pseudoframes for affine subspaces by means of the Beurling dimension of their sets of parameters. This provides us, in particular, with a Nyquist dimension which separates sets of parameters of pseudoframes from those of non-pseudoframes and which links a fixed value to sets of parameters of pseudo-Riesz sequences. These results are even new for the special case of Gabor frames for an affine subspace.      

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?

     

The A-integral and restricted Ahlfors–Beurling transform

In this paper we prove that the restricted Ahlfors–Beurling transform of a Lebesgue integrable function is A-integrable and derive an analogue of Riesz's equality holds.     

Properties of the Beurling generalized primes

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In this paper, we prove a generalization of Mertens' theorem to Beurling primes, namely that , where γ is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit exists. We also show that this limit coincides with ; for ordinary primes this claim is called Meissel's theorem. Finally, we will discuss a problem posed by Beurling, namely how small |N(x)−[x]| can be made for a Beurling prime number system Q≠P, where P is the rational primes. We prove that for each c>0 there exists a Q such that and conjecture that this is the best possible bound.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=Kw3iNo3fAbk/.     

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.     

Characterization of tail distributions based on record values by using the Beurling’s Tauberian theorem

In this paper, we mainly investigate the converse of a well-known theorem proved by Shorrock (J. Appl. Prob. 9, 316–326 1972b), which states that the regular variation of tail distribution implies a non-degenerate limit for the ratios of the record values. Specifically, the converse is proved by using Beurling extension of Wiener’s Tauberian theorem. This equivalence is extended to the Weibull and Gumbel max-domains of attraction.     

On the distribution of Beurling integers

Asymptotic behaviour of the counting function of Beurling integers is deduced from Chebyshev upper bound and Mertens formula for Beurling primes. The proof based on some properties of corresponding zeta-function on the right of its abscissa of convergence.     

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 .     

Uncertainty Principles for the Generalized Fourier Transform Associated with Spherical Mean Operator

The aim of this paper is to establish an extension of qualitative and quantitative uncertainty principles for the Fourier transform connected with the spherical mean operator.     

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.