Sequences of composition operators in spaces of functions of bounded Φ-variation

The main results of the paper are contained in Theorems 1 and 2. Theorem 1 presents necessary and sufficient conditions for a sequence of functions h _n : 〈c, d〉 → 〈a, b〉, n = 1, 2, ..., to have bounded sequences of Ψ-variations {V _Ψ (〈c, d〉; f ? h _n )} _n=1 ^∞ evaluated for the compositions of an arbitrary function f: 〈a, b〉 → ? with finite Φ-variation and the functions h _n . In Theorem 2, the same is done for a sequence of functions h _n : ? → ?, n = 1, 2, ..., and the sequence of Ψ-variations {V _Ψ(〈a, b〉; h _n ? f)} _n=1 ^∞ .     

On Vector Valued Measure Spaces of Bounded Φ-variation containing copies of ℓ∞

Given a Young function , we study the existence of copies of c ₀ and in cabv (,X) and in cabsv (,X), the countably additive, -continuous, and X-valued measure spaces of bounded -variation and bounded -semivariation, respectively.     

Subseries convergence in the space of functions of bounded φ-variation

Let there be given two F-spaces (X, ∥ ∥*), (Y, ∥ ∥*), X ? Y. A series Σ ₁ ^∞ x_i of elements in X is said to have property O(X, Y) (in the case when X = Y, ∥ ∥ = ∥ ∥*-property O) if perfect boundedness of this series in (X, ∥ ∥) implies its perfect (subseries) convergence in ( Y, ∥ ∥*). Conditions are determined for a series of elements in the space of functions of bounded φ-variation to have property O(X, Y). These conditions are rephrased for the case when convergence with respect to the norm ∥∥ _? ^v is replaced by modular convergence generated by φ-variation.     

On zeros of polynomials of best approximation to holomorphic andC ∞ functions

LetE be a compact subset of the complex planeC such that Leja's extremal functionL _E forE is continuous. If almost all zeros of the polynomials of best approximation to a functionfC(E) are outside the setE _R ={zC:L _E (z<R)}, for someR>1, thenf is extendible to a holomorphic function inE _R . If the zeros ofn-th, polynomial of best approximation tof are outside and the sequence {R _n ^–n } rapidly decreases to zero thenf can be extended to aC function on 075-4}.     

On groups of type Φ

We define a group G to be of type Φ if it has the property that for every -module G, proj. G < ∞ iff proj. H G < ∞ for every finite subgroup H of G. We conjecture that the type Φ is an algebraic characterization of those groups G which admit a finite dimensional model for , the classifying space for the family of the finite subgroups of G. We also conjecture that the type Φ is equivalent to spli being finite, where spli is the supremum of the projective lengths of the injective -modules. Here we prove certain parts of these conjectures. The project is cofounded by the European Social Fund and National Resources–EPEAK II–Pythagoras. Received: 21 June 2006     

On Φ-Derivation in Banach Algebras

Using Bre?ar and ?emrl approach, we give a proof of the extended Jacobson density theorem for Φ-derivations. Further, some applications on Banach algebras will be given. Precisely, for d being a continuous Φ-derivation on a given Banach algebra ?, we show that: d(?) ? rad (?) ? [b,[a, d(a)]] ∈ rad (?) for all a, b ∈ ? and d leaves invariant all maximal ideals of codimension one?for every a ∈ ? there exists a positive integer n such that (d(a)) ⁿ is quasi-nilpotent ? [d, Φ](?) ? rad (?) and d ²(a) ∈ rad (?) for all a ∈ ?. Finally, we characterize all pairs d, δ of continuous Φ-derivations such that dδ(a) is quasi-nilpotent for all a ∈ ? and [d, δ](?), [d, Φ](?), [δ,Φ](?) are subsets of rad (?).     

Fourier coefficients of functions from many-dimensional classes of bounded Λ-variation

Estimates of trigonometric Fourier coefficients are obtained for functions from the classes of bounded many-dimensional Λ-variation and functions continuous in Λ-variation. In some cases it is proved that these estimates are unimprovable by the order.     

Almost periodic functions in terms of Bohr’s equivalence relation

In this paper we introduce an equivalence relation on the classes of almost periodic functions of a real or complex variable which is used to refine Bochner’s result that characterizes these spaces of functions. In fact, with respect to the topology of uniform convergence, we prove that the limit points of the family of translates of an almost periodic function are precisely the functions which are equivalent to it, which leads us to a characterization of almost periodicity. In particular we show that any exponential sum which is equivalent to the Riemann zeta function, \(\zeta (s)\), can be uniformly approximated in \(\{s=\sigma +it:\sigma >1\}\) by certain vertical translates of \(\zeta (s)\).     

Examples of divergent fourier series for classes of functions of bounded Λ-variation

The author has shown earlier that the requirement that a continuous function belong to the class HBV ([?π, π] ^m ) for m ≥ 3 is not sufficient for the convergence of its Fourier series over rectangles. The author gave examples of functions of three and more variables from the Waterman class which are harmonic in the first variable and significantly narrower in the other variables and whose Fourier series are divergent at some point even on cubes. In the present paper, this assertion is strengthened. The main result is that such an example can be constructed even when the class with respect to the first variable is somewhat narrowed. Also the one-dimensional result due to Waterman is refined.     

On the approximation of conjugate functions from Holder classes by Fejér means

Получены асимптотич еские равенства для в еличин гдеr≧0 — целое, ω(t) — выпу клый модуль непрерыв ности и $$\bar \sigma _n (f;x) = - \frac{1}{\pi } \mathop \smallint \limits_{ - \pi }^\pi f(x + t)\left( {\frac{1}{2}ctg\frac{t}{2} - \frac{1}{{4(n + 1)}}\frac{{\sin (n + 1)t}}{{\sin ^2 \tfrac{1}{2}t}}} \right)dt$$ сумма Фейера функцииf(х), сопряженной сf(x).     

Localization of Cesaro means of Fourier series for functions of bounded Λ-variation

We consider classes of periodic functions of bounded Λ-variation, where Λ has a power growth rate. We show that this class contains a continuous function whose Cesaro means of the Fourier series (whose order depends on the growth rate of Λ) have no localization property.     

On the approximation of periodic functions by de la Vallée Poussin sums

Пустьf — непрерывная периодическая функц ия,s _n (f) — сумма Фурье порядкаn функцииf,E _n (f) — наилучшее прибли жениеf тригонометри ческими полиномами порядкаn в чебьппев-ской метрике и $$\sigma _{n, m} (f) = \frac{1}{{m + 1}}\mathop \sum \limits_{v = n - m}^n s_v (f) (0 \leqq m \leqq n; n = 0, 1, \ldots )$$ — суммы Bалле Пуссена ф ункцииf Для любой последовательностиε={ε_v} (v=0, l,...),ε _v ↓0(v→∞) обозначим чер езC(ε) класс непрерывн ых функцийf, для которыхE _v (f)≦ε _v (v=0,1,...). В работе устанавли вается, что существую т абсолютные положите льные кон-стантыa ₁ иa ₂ такие, что $$A_1 \mathop \sum \limits_{v = 0}^n \frac{{\varepsilon _{n - m + v} }}{{m + v + 1}} \leqq \mathop {\sup }\limits_{f \in C(\varepsilon )} \parallel f - \sigma _{n, m} (f)\parallel \leqq A_2 \mathop \sum \limits_{v = 0}^n \frac{{\varepsilon _{n - m + v} }}{{m + v + 1}}$$ для всех 0≦m≦n; n=0, l, ... В частн ых случаяхт=п иm=0 этот результат равноси-ле н теоремам, установлен ным ранее автором и К. И. Осколковым.     

On σ-commutativity in Menger algebras of n-place functions

We present an abstract characterization of various types of Menger algebras of n-place functions allowing certain permutations of variables.     

Schrödinger potentials solvable in terms of the confluent Heun functions

We show that if the potential is proportional to an energy-independent continuous parameter, then there exist 15 choices for the coordinate transformation that provide energy-independent potentials whose shape is independent of that parameter and for which the one-dimensional stationary Schrödinger equation is solvable in terms of the confluent Heun functions. All these potentials are also energy-independent and are determined by seven parameters. Because the confluent Heun equation is symmetric under transposition of its regular singularities, only nine of these potentials are independent. Five of the independent potentials are different generalizations of either a hypergeometric or a confluent hypergeometric classical potential, one potential as special cases includes potentials of two hypergeometric types (the Morse confluent hypergeometric and the Eckart hypergeometric potentials), and the remaining three potentials include five-parameter conditionally integrable confluent hypergeometric potentials. Not one of the confluent Heun potentials, generally speaking, can be transformed into any other by a parameter choice.     

On some extremal problems in the theory of approximation of functions in the spaces S p , 1 ≤ p < ∞

We consider and study properties of the smoothness characteristics , of functions f(x) that belong to the space S ^p , 1 ≤ p < ∞, introduced by Stepanets. Exact inequalities of the Jackson type are obtained, and the exact values of the widths of the classes of functions defined by using Ω_m(f,t)_S ^p are calculated. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 303–316, March, 2006.     

ϕ-Strong approximation of functions by trigonometric polynomials

Mathematical Notes - The rate of ?-strong approximation of periodic functions by trigonometric polynomials constructed on the basis of interpolating polynomials with equidistant nodes is...     

On the growth and approximation of entire functions represented by Laplace–Stieltjes’ transformation

In the present paper, we obtain the characterization of various growth parameters of an entire function F(s) represented by Laplace–Stieltjes transformation in terms of the rate of decrease of \(E_n ( {F,\beta } ),\)where \(E_n ( {F,\beta } )\) represents the error in approximating the function F(s) by exponential polynomials.     

On finite element approximation in theL ∞-norm of variational inequalities

Summary We are interested in the approximation in theL -norm of variational inequalities with non-linear operators and somewhat irregular obstacles. We show that the order of convergence will be the same as that of the equation associated with the non-linear operator if the discrete maximum principle is verified.