A study on weighted mean summability

In the present paper, a general theorem on summability factors of infinite series has been proved. Also we have obtained a new result concerning the |C,1;δ| _k summability factors.     

A study on absolute Riesz summability factors

In the present paper, two general theorems on summability factors of infinite series which generalize some known results, have been proved. Also we have obtained two new results concerning the |C,1;δ| _k summability factors.     

Absolute summability of infinite series

It is shown in [4] that if a normal matrix,A satisfies some conditions then |C,1| _k summability implies |A| _k summability wherek≥1. In the present paper, we consider the converse implication.     

Factors for ¦N,pn;δ¦k summability of Fourier series

In this paper two theorems on | N,p_n;δ|_k summability factors, which generalize the results of Bor [4] on | N,p_n|k summability factors, have been proved.     

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.     

A note on two absolute summability methods

In this paper we have established a relation between |R,p _n ;δ|_k and |R,q _n ;δ|_k summability methods which generalizes the results of Bor [1] and Bosanquet [2].     

On the Absolute Matrix Summability Factors of Fourier Series

In this paper, two known theorems on |N?, p _n | _k summability methods of Fourier series have been generalized for |A, p _n | _k summability factors of Fourier series by using different matrix transformations. New results have been obtained dealing with some other summability methods.

     

A general theorem characterizing some absolute summability methods

A general theorem is given which gives the necessary and sufficient conditions satisfied by a sequence (ε_n) in order to have the series Σa _n ε _n summable to |A| whenever Σa _n is summable to |A| for some summability methodA.     

General absolute summability factor theorems involving quasi-power-increasing sequences

In this paper a general theorem concerning the |A,δ|_k summability methods has been proved, which generalizes two results of Şevli and Leindler [H. Şevli and L. Leindler, On the absolute summability factors of infinite series involving quasi-power-increasing sequences, Computers and Mathematics with Applications 57 (2009), 702–709]. We obtain sufficient conditions for ∑a_nλ_n to be summable |A,δ|_k, k1, 0δ<1/k, by using quasi-power-increasing sequences.     

The Subadditive Ergodic Theorem and generic stretching factors for free group automorphisms

Let F _k be a free group of rank k ≥ 2 with a fixed set of free generators. We associate to any homomorphism φ from F _k to a group G with a left-invariant semi-norm a generic stretching factor, λ(φ), which is a noncommutative generalization of the translation number. We concentrate on the situation where φ: F _k → Aut(X) corresponds to a free action of F _k on a simplicial tree X, in particular, where φ corresponds to the action of F _k on its Cayley graph via an automorphism of F _k . In this case we are able to obtain some detailed “arithmetic” information about the possible values of λ = λ(φ). We show that λ ≥ 1 and is a rational number with 2kλ ∈ ℤ[1/(2k − 1)] for every φ ∈ Aut(F _k ). We also prove that the set of all λ(φ), where φ varies over Aut(F _k ), has a gap between 1 and 1+(2k−3)/(2k ²−k), and the value 1 is attained only for “trivial” reasons. Furthermore, there is an algorithm which, when given φ, calculates λ(φ). The second and the third author were supported by the NSF grant DMS#0404991 and the NSA grant DMA#H98230-04-1-0115.     

Minimum nastiness curve fitting

The ‘nastiness’ of a function φ(x) is defined. We then discuss minimum nastiness interpolation to a set of given points (x _k, φ_k), as well as minimum nastiness curve fitting, where the given valuesφ _k have errorsδ _k.     

A new factor theorem for absolute matrix summability

Abstract

In this paper, we have extended a known theorem dealing with |, p_n|k summability factors to the |A, p_n|k summability under weaker conditions by using an almost increasing sequence     

On a Class of Vector-Valued Sequences Associated with Multiplier Sequences

In this article we introduce the vector valued sequence space m(E_k,φ,∧),associated with themultiplier sequence ∧=(λ_k) of non-zero complex numbers,and the terms of the sequence are chosen from theseminormed spaces E_k,seminormed by f_k for all k∈N.This generalizes the sequence space m(φ) introducedand studied by Sargent.We study some of its properties like solidity,completeness,and obtain some inclusionresults.We also characterize the multiplier problem and obtain the corresponding spaces dual to m(E_k,φ,∧).We prove some general results too.     

Generalized absolutely monotone functions

The concept of absolutely monotone functions is generalized by replacing the conditionsφ ^(k)(t)≧0,k=0, 1, … by an infinite sequence of differential inequalitiesφ(t)≧0,L _kφ(t)≧0,k=1, 2, …, where theL _k are differential operators of a special type. It is shown that these functions have a valid series expansion in terms of basic functions associated with the operatorsL _k.     

Generalized difference sequence s pace defined by |$$ \bar N $$, pk| summability and an Orlicz function in seminormed s pace

The object of this paper is to introduce a new difference sequence space which arise from the notions of |$ \bar N $ \bar N , p _k | summability and an Orlicz function in seminormed complex linear space. Various algebraic and topological properties and certain inclusion relations involving this space have been discussed. This study generalizes results: [ALTIN, Y.—ET, M.—TRIPATHY, B. C.: The sequence space |$ \bar N_p $ \bar N_p |(M, r, q, s) on seminormed spaces, Appl. Math. Comput. 154 (2004), 423–430], [BHARDWAJ, V. K.—SINGH, N.: Some sequence spaces defined by |$ \bar N $ \bar N , p _n | summability, Demonstratio Math. 32 (1999), 539–546] and [BHARDWAJ, V. K.—SINGH, N.: Some sequence spaces defined by |$ \bar N $ \bar N , p _n | summability and an Orlicz function, Indian J. Pure Appl. Math. 31 (2000), 319–325].     

Regularity of refinable function vectors

We study the existence and regularity of compactly supported solutions φ = (φ_v) _v=0 ^/r−1 of vector refinement equations. The space spanned by the translates of φ_v can only provide approximation order if the refinement maskP has certain particular factorization properties. We show, how the factorization ofP can lead to decay of |̸_v(u)| as |u| → ∞. The results on decay are used to prove uniqueness of solutions and convergence of the cascade algorithm.     

A characteristic property of spheres

Summary We prove: Let S be a closed n-dimensional surface in an(n+1)-space of constant curvature (n ≥ 2); k₁ ≥ ... ≥ k_n denote its principle curvatures. Let φ(ξ₁, ..., ξ_n) be such that . Then if φ(k₁, ..., k_n)=const on S and S is subject to some additional general conditions (those(II ₀) or(II) n^o 1), S is a sphere. To Enrico Bompiani on his scientific Jubilee     

On the Absolute Summability of Series with Respect to Block-Orthonormal Systems

Theorems determining Weyl's multipliers for the summability almost everywhere by the |c, 1| method of the series with respect to block-orthonormal systems are proved. In particular, it is stated that if the sequence {(n)} is the Weyl multiplier for the summability almost everywhere by the |c, 1| method of all orthogonal series, then there exists a sequence {N _k} such that {(n)} will be the Weyl multiplier for the summability almost everywhere by the |c, 1| method of all series with respect to the _k -orthonormal systems.     

The Gleason’s problem for some polyharmonic and hyperbolic harmonic function spaces

Let Ω ⊆ ℝⁿ be a bounded convex domain with C ² boundary. For 0 < p, q ⩽ ∞ and a normal weight φ, the mixed norm space H _k ^p,q,φ (Ω) consists of all polyharmonic functions f of order k for which the mixed norm ∥ · ∥_p,q,φ < ∞. In this paper, we prove that the Gleason’s problem (Ω, a, H _k ^p,q,φ ) is always solvable for any reference point a ∈ Ω. Also, the Gleason’s problem for the polyharmonic φ-Bloch (little φ-Bloch) space is solvable. The parallel results for the hyperbolic harmonic mixed norm space are obtained.     

Groups whose nonlinear irreducible characters separate element orders or conjugacy class sizes

A class function φ on a finite group G is said to be an order separator if, for every x and y in G \ {1}, φ(x) = φ(y) is equivalent to x and y being of the same order. Similarly, φ is said to be a class-size separator if, for every x and y in G\ {1}, φ(x) = φ(y) is equivalent to |C _G (x)| = |C _G (y)|. In this paper, finite groups whose nonlinear irreducible complex characters are all order separators (respectively, class-size separators) are classified. In fact, a more general setting is studied, from which these classifications follow. This analysis has some connections with the study of finite groups such that every two elements lying in distinct conjugacy classes have distinct orders, or, respectively, in which disctinct conjugacy classes have distinct sizes. Received: 10 April 2007