On absolute summability factors of infinite series

In this paper using δ-quasi-monotone sequences a theorem on summability factors of infinite series, which generalizes a theorem of Bor [4] on summability factors of infinite series, is proved. Also, in the special case this theorem includes a result of Mazhar [8] on |C, 1|_k summability factors.     

Accelerating infinite products

Slowly convergent infinite products are considered, where is a sequence of numbers, or a sequence of linear operators. Using an asymptotic expansion for the “remainder” of the infinite product a method for convergence acceleration is suggested. The method is in the spirit of the d-transformation for series. It is very simple and efficient for some classes of sequences . For complicated sequences it involves the solution of some linear systems, but it is still effective. This revised version was published online in June 2006 with corrections to the Cover Date.     

Necessary and sufficient conditions for inclusion relations for absolute summability

We obtain a set of necessary and sufficient conditions for to imply for 1 <k ≤ s < ∞. Using this result we establish several inclusion theorems as well as conditions for the equivalence of and .     

A note on absolute summability factors

Summary We obtain sufficient conditions for the series <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\sum a_{n}\lambda_{n}$ to be absolutely summable of order $k$ by a<span lang=FR style='font-size:10.0pt; mso-ansi-language:FR'>triangular matrix.     

Nil-algebras and infinite groups

We simplify our construction of nil-algebras by proving, for any integer d ≥ 2 and over any field , that there exists a residually nilpotent, nonnilpotent nil-algebra over generated by d elements. As a consequence, we obtain similar results for nonassociative algebras and groups. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 189–200, 2005.     

Pointwise strong and very strong approximation of Fourier series

We present an estimation of the and H _u ^λφ f means as approximation versions of the Totik type generalization (see [6, 7]) of the result of G. H. Hardy, J. E. Littlewood, considered by N. L. Pachulia in [5]. Some results on the norm approximation will also be given.      

On some new criteria for infinite differentiability of periodic functions

We study the set of infinitely differentiable periodic functions in terms of generalized -derivatives defined by a pair of sequences ψ ₁ and ψ ₂. It is shown that every function ƒ from the set has at least one derivative whose parameters ψ ₁ and ψ ₂ decrease faster than any power function and, at the same time, for any function ƒ ∈ different from a trigonometric polynomial, there exists a pair ψ whose parameters ψ ₁ and ψ ₂ have the same rate of decrease and for which the -derivative no longer exists. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1399–1409, October, 2007.     

Weak infinite powers of Blaschke products

Letb be a Blaschke product with zeros {z _n } in the open unit disk Δ. Let be the set of sequences of non-negative integersp=(p ₁,p ₂,…) such that ∑ _n=1 ^∞ p _n (1 − |z _n |) < ∞ andp _n →∞ asn→∞. We study the class of weak infinite powers ofb, Properties of these classes depend on the setS(b) of the cluster points in ∂Δ of {z _n }. It is proved thatS(b)=∂Δ if and only if , the Douglas algebra generated by . Also, it is proved thatdθ(S(b))=0 if and only if there exists an interpolating Blaschke productB such that .     

Differential equations of infinite order for orthogonal polynomials

Summary Differential equations of infinite order of the form , where M_k is a polynomial of degree ≦k, have polynomial solutions for suitable values of λ. Necessary and sufficient conditions are given in order that these solutions form a set of orthogonal polynomials. The cases where { degree M_k }=1 and 2 are studied in detail. Supported by N.S.F. Grant GP-5311.     

Borel hierarchies in infinite products of Polish spaces

Let H be a product of countably infinite number of copies of an uncountable Polish space X. Let Σ_ξ be the class of Borel sets of additive class ξ for the product of copies of the discrete topology on X (the Polish topology on X), and let . We prove in the Lévy-Solovay model that

Solutions of multiple vector refinement equations with infinite mask

This paper is devoted to investigating the solutions of refinement equations of the form Ф（x）=∑α∈Z^s α（α）Ф（Mx-α）,x∈R^s,where the vector of functions Ф = （Ф1,… ,Фr）^T is in （L1（R^s））^r, α =（α（α））α∈Z^s is an infinitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim n→∞ M^-n =0, with m = detM. Some properties about the solutions of refinement equations axe obtained.     

Evaluation of infinite series involving special products and their algebraic characterization

The aim of the paper is the investigation of special infinite series of the form

Resolvability of infinite designs

In this paper we examine the resolvability of infinite designs. We show that in stark contrast to the finite case, resolvability for infinite designs is fairly commonplace. We prove that every t  -(v,k,Λ)$(v,k,\Lambda )$ design with t finite, v   infinite and k,λ<v$k,\lambda <v$ is resolvable and, in fact, has α   orthogonal resolutions for each α<v$\alpha <v$. We also show that, while a t  -(v,k,Λ)$(v,k,\Lambda )$ design with t and λ finite, v   infinite and k=v$k=v$ may or may not have a resolution, any resolution of such a design must have v parallel classes containing v   blocks and at most λ−1$\lambda -1$ parallel classes containing fewer than v   blocks. Further, a resolution into parallel classes of any specified sizes obeying these conditions is realisable in some design. When k<v$k<v$ and λ=v$\lambda =v$ and when k=v$k=v$ and λ is infinite, we give various examples of resolvable and non-resolvable t  -(v,k,Λ)$(v,k,\Lambda )$ designs.     

On the equivalence of differential operators of infinite order in analytic spaces

It is shown that in the spacesA _R (0 <R ⩽ ∞) of all functions which are single-valued and analytic in the disk |z| < R with the topology of compact convergence, the differential operator of infinite order with constant coefficients is equivalent to the operator Dⁿ (n is a fixed natural number) if and only if and |ϕ _n | = 1 for R < ∞ or ϕ _n ≠ 0 for R = ∞. Also the equivalence of two shift operators in the space A_∞ is investigated. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 33–37, January, 1977.     

Strong Cesàro summability of double Fourier integrals

We prove the following theorem. Assume f ∈ L ^∞(R ²) with bounded support. If f is continuous at some point (x ₁,x ₂) ∈ R ², then the double Fourier integral of f is strongly q-Cesàro summable at (x ₁,x ₂) to the function value f(x ₁,x ₂) for every 0 < q < ∞. Furthermore, if f is continuous on some open subset of R ², then the strong q-Cesàro summability of the double Fourier integral of f is locally uniform on . Research partially supported by the Australian Research Council and the Hungarian National Foundation for Scientific Research under Grant T 046 192.     

A note on summability of multiple Laguerre expansions

A simple structure of the multiple Laguerre polynomial expansions is used to study the Cesàro summability above the critical index for the convolution type Laguerre expansions. The multiple Laguerre polynomial expansion of an -radial function is shown to be an -radial function that coincides with the Laguerre polynomial expansion of , which allows us to settle the problem of summability below the critical index for the -radial functions.

     

Finite and infinite systems of interacting diffusions

Summary We study the problem of relating the long time behavior of finite and infinite systems of locally interacting components. We consider in detail a class of lincarly interacting diffusionsx(t)={x _i (t),i ∈ ℤ ^d } in the regime where there is a one-parameter family of nontrivial invariant measures. For these systems there are naturally defined corresponding finite systems, , with . Our main result gives a comparison between the laws ofx(t _N ) andx ^N (t _N ) for timest _N →∞ asN→∞. The comparison involves certain mixtures of the invariant measures for the infinite system. Partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University, by the National Science Foundation, and by the National Security Agency Research supported in part by the DFG Partly supported by S.R.63540155 of Japan Ministry of Education     

An extension of the integral test

We identify a class of functions for which an extension of the well known integral test holds, namely, the series is Cesàro summable if and only if the integral is. We also give the corresponding multidimensional results.

     

Sojourn times,manifolds with infinite cylindrical ends,and an inverse problem for planar waveguides

We prove that two particular entries in the scattering matrix for the Dirichlet Laplacian on ℝ × (−γ, γ) determine an analytic strictly convex obstacle . With an additional symmetry assumption, one entry suffices. Part of the proof is an integral identity involving an entry in the scattering matrix and a distribution related to the fundamental solution of the wave equation. This identity holds for general manifolds with infinite cylindrical ends. A consequence of this is a relationship between the singularities of the Fourier transform of an entry in the scattering matrix and the sojourn times of certain geodesics. Partially supported by NSF grant DMS 0500267.     

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.