A continuous extension of the de la Vallée Poussin means

Among the many interesting results of their 1958 paper, G. Pólya and I. J. Schoenberg studied the de la Vallée Poussin means of analytic functions. These are polynomial approximations of a given analytic function on the unit disk obtained by taking Hadamard products of the functionf with certain polynomialsV _n (z), wheren is the degree of the polynomial. The polynomial approximationsV _n *f converge locally uniformly tof asn→∞. In this paper, we define a subordination chainV _λ (z),γ>0, |z|<1, of convex mappings of the disk that for integer values is the same as the previously definedV _n (z). Iff is a conformal mapping of the diskD onto a convex domain, thenV _λ *f→f locally uniformly as λ→∞, and in fact when λ₂ > λ₁. We also consider Hadamard products of theV _λ with complex-valued harmonic mappings of the disk. This work was supported by the Volkswagen Stiftung (RiP-program at Oberwolfach). S. R. received partial support also from INTAS (Project 99-00089) and the German-Israeli Foundation (grant G-643-117.6/1999).     

A new approach to the Assad–Kirk fixed point theorem

In the present paper, considering the Wardowski’s technique, we give a new approach to the Assad–Kirk fixed point theorem on metrically convex metric spaces.We also provide a nontrivial example showing that our result is a proper extension of the Assad–Kirk fixed point theorem.     

On the divergence of de la Vallée Poussin means of Fourier series

Пусть {λ _{n 1} ^t8 — монотонн ая последовательнос ть натуральных чисел. Дл я каждой функции fεL(0, 2π) с рядом Фурье строятся обобщенные средние Bалле Пуссена $$V_n^{(\lambda )} (f;x) = \frac{{a_0 }}{2} + \mathop \sum \limits_{k = 1}^n (a_k \cos kx + b_k \sin kx) + \mathop \sum \limits_{k = n + 1}^{n + \lambda _n } \left( {1 - \frac{{k - n}}{{\lambda _n + 1}}} \right)\left( {a_k \cos kx + b_k \sin kx} \right).$$ Доказываются следую щие теоремы.

1. Если λ_n=o(n), то существуе т функция fεL(0, 2π), для кот орой последовательность {V_n ^(λ)(?;x)} расходится почти вс юду.
2. Если λ_n=o(n), то существуе т функция fεL(0, 2π), для кот орой последовательность $$\left\{ {\frac{1}{\pi }\mathop \smallint \limits_{ - \pi /\lambda _n }^{\pi /\lambda _n } f(x + t)\frac{{\sin (n + \tfrac{1}{2})t}}{{2\sin \tfrac{1}{2}t}}dt} \right\}$$ расходится почти всю ду

.     

A new algebraic approach to the Szegő-Widom limit theorem

We give another proof of the Szeg\H{o}–Widom Limit Theorem. This proof relies on a new Banach algebra method that can be directly applied to the asymptotic computation of the Toeplitz determinants. As a by-product, we establish an interesting identity for operator determinants of Toeplitz operators, namely if are certain matrix valued functions defined on the unit circle, then

On the degree of approximation by the operators of de la Vallée Poussin

In this paper we study the degree of approximation of functionsf inC ₂ andC ₂ ¹ by the operatorsV _n ofde la Vallée Poussin. The quality of approximation is measured in terms of the modulus of continuity off andf respectively. Forn so-called exact constants of approximation are determined. Furthermore, the asymptotic behaviour of these constants is investigated asn.     

A nonlinear analytic approach to the splitting theorem of Jacobs–de Leeuw–Glicksberg

In this paper, we study nonlinear analytic methods for linear contractive semigroups in Banach spaces and apply them to the splitting theorem of Jacobs–de Leeuw–Glicksberg. Using these results, we obtain the extension of Lin’s proposition for a group of linear operators to a semigroup.     

A new topological proof of the Abhyankar–Moh theorem

We present a new simple proof of the famous theorem of Abhyankar, Moh and Suzuki about rational curves in a plane. This proof relies on the Poincaré–Hopf theorem. This work was supported by Polish KBN Grant No 2 P03A 041 15Mathematics Subject Classification (2002):14E25.     

Controllability of the Vallée–Poussin Problem for Impulsive Differential Systems

Necessary and sufficient conditions for the controllability of multipoint boundary-value problems for linear semihomogeneous and nonhomogeneous impulsive differential systems are established. The set of all controls solving such problems is described. Moreover, we provide sufficient conditions for the time-optimal control of the Vallée–Poussin problem and obtain the Pontryagin maximum principle in sufficient form.     

Approximations of functions defined on the real axis by means of de la Vallée-Poussin operators

Asymptotic equations for upper bounds of deviations of the de la Vallée-Poussin operators on C classes in the uniform metric are obtained.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 5, pp. 682–691, May, 1992.     

A new approach to the stabilization of a fluid–structure interaction model

This article is a continuation of our work on a linear fluid–structure interaction model [Grobbelaar-Van Dalsen, On a fluid–structure model in which the dynamics of the structure involves the shear stress due to the fluid, J. Math. Fluid Mech. 10(3) (2008), pp. 388–401; Grobbelaar-Van Dalsen, Strong stability for a fluid––structure model, Math. Methods Appl. Sci., 32(2009) pp. 1452–1466]. The model describes the interaction between a 3-D incompressible fluid and a 2-D plate, the interface, which coincides with a flat flexible part of the surface of the vessel containing the fluid. The mathematical model comprises the Stokes equations and the equations for the longitudinal deflections of the plate with the inclusion of the shear stress that the fluid exerts on the plate. A dissipative damping mechanism of Kelvin–Voigt type is applied to the interior of the plate. While our earlier work shows that weak solutions in a space of finite energy are strongly asymptotically stable under no-slip transmission conditions at the interface with uniform exponential stability only attainable under an additional domination condition, the present research is directed at achieving uniform exponential stability of weak solutions without imposing the domination condition. Using energy methods we establish uniform exponential decay under a modified transmission condition at the interface. This condition entails that the fluid velocity at the interface is coupled to a linear combination of the plate velocity and displacement.     

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.     

Asymptotics of the approximation of continuous and differentiable functions by the singular integrals of de la Vallée Poussin

The complete asymptotic developments in powers of 1/n are derived for quantities characterizing approximation by singular integrals of de la Vallée Poussin $$V_n (f:x) = \frac{1}{{\Delta _n }}\int_{ - \pi }^\pi {f(x + t)} \cos ^{2n} \frac{t}{2}dt;\Delta _n = \int_{ - \pi }^\pi {\cos ^{2n} \frac{t}{2}dt}$$ of the function classes Lipa, 0w ^(r), r?1 an integer.     

A subjective Bayesian approach to the theory of queues I — Modeling

This is the first of an expository two-part paper which outlines a point of view different from that currently used in queueing theory. In both parts, the focus is on concepts. Here we adopt a personal probability point of view to all sources of uncertainty in the theory of queues and explore the consequences of our approach by comparing our results to those that are currently available. For ease of exposition, we confine attention to the M/M/1/ and the M/M/1/K queues. In Part I we outline the general strategy and focus on model development. In Part II we address the problem of inference in queues within the subjective Bayesian paradigm and introduce a use of Shannon's measure of information for assessing the amount of information conveyed by the various types of data from queues.     

A new combinatorial approach to the construction of constant composition codes

Constant composition codes(CCCs)are a new generalization of binary constant weight codes and have attracted recent interest due to their numerous applications. In this paper, a new combinatorial approach to the construction of CCCs is proposed, and used to establish new optimal CCCs.     

Nielsen theory, Floer homology and a generalisation of the Poincaré-Birkhoff theorem

The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology for symplectomorphisms of surfaces and a calculation of Seidel’s symplectic Floer homology for different mapping classes. We also describe symplectic zeta functions and an asymptotic symplectic invariant. A generalisation of the Poincaré-Birkhoff fixed point theorem and Arnold conjecture is proposed. Dedicated to Vladimir Igorevich Arnold     

A new class of generalized Apostol–Bernoulli polynomials and some analogues of the Srivastava–Pintér addition theorem

The main purpose of this paper is to introduce and investigate a new class of generalized Apostol–Bernoulli polynomials based on a definition given by Natalini and Bernardini (2003) [22] for the generalized Bernoulli polynomials. We obtain a generalization of the Srivastava–Pintér addition theorem (Srivastava and Pintér (2004) [23]). We also give a list of expressions involving special functions that could be used to obtain some analogues of the Srivastava–Pintér addition theorem. Finally, we give an analogue featuring the new class of generalized Apostol–Bernoulli polynomials.