Vallee Poussin Research Articles

Approximation of Continuous Functions by de La Vallee-Poussin Operators

For the upper bounds of the deviations of a function defined on the entire real line from the corresponding values of the de la Vallee-Poussin operators, we find asymptotic equalities that give a solution of the well-known Kolmogorov–Nikol'skii problem.

Interpolating polynomial wavelets on [?1,1

The present paper gives a contribution of wavelet aspects to classical algebraic polynomial approximation theory. As is so often the case in classical approximation, the authors follow the pattern provided by the trigonometric polynomial case. Algebraic polynomial interpolating scaling functions and wavelets are constructed by using the interpolation properties of de la Vallee Poussin kernels with respect to the four kinds of Chebyshev weights. For the decomposition and reconstruction of a given function the structure of the involved matrices is studied in order to reduce the computational effort by means of fast discrete cosine and sine transforms.

Approximation by Means of h-Harmonic Polynomials on the Unit Sphere

The h-harmonics are analogues of the ordinary harmonics, they are orthogonal homogeneous polynomials on the sphere with respect to a weight function that is invariant under a reflection group. Two means of associated orthogonal expansions, the de la Vallee Poussin means and an analog of spherical means, are defined and their approximation behaviors are studied. A weighted modulus of smoothness is defined using the modified spherical means and is proved to be equivalent to a weighted K-modulus defined using the differential-difference h-spherical Laplacian. A Bernstein type inequality for the h-spherical Laplacian is also established.

On summability of weighted Lagrange interpolation. II

Starting from the Lagrange interpolation on the roots of Jacobi polynomials, a wide class of discrete linear processes is constructed using summations. Some special cases are also considered, such as the Fejer, de la Vallee Poussin, Cesaro, Riesz and Rogosinski summations. The aim of this note is to show that the sequences of this type of polynomials are uniformly convergent on the whole interval [-1,1] in suitable weighted spaces of continuous functions. Order of convergence will also be investigated. Some statements of this paper can be obtained as corollaries of our general results proved in [15].

-summability of Fourier series

A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions on Θ we show that if the maximal Fejer operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means of these Fourier series is bounded from H p to L p (1/2<p≤; ∞) and is of weak type (1,1). In the endpoint case p=1/2 a weak type inequality is derived. As a consequence we obtain that the Θ-means of a function f∈L 1 converge a.e. to f. Some special cases of the Θ-summation are considered, such as the Weierstrass, Picar, Bessel, Riesz, de la Vallee-Poussin, Rogosinski and Riemann summations. Similar results are verified for several-dimensional Fourier series and Hardy spaces.

Approximation of Poisson Integrals by de la Vallée Poussin Sums

On the classes of Poisson integrals of functions belonging to unit balls in the spaces C and L, we obtain asymptotic equalities for the upper bounds of approximations by de la Vallee Poussin sums in the uniform metric and the integral metric, respectively.

A continuous extension of the de la Vallée Poussin means

Among the many interesting results of their 1958 paper, G. Polya and I. J. Schoenberg studied the de la Vallee 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 $$V_{\lambda _1 } * f(\mathbb{D}) \subset V_{\lambda _2 } * f(\mathbb{D}) \subset f(\mathbb{D})$$ when λ2 > λ1. We also consider Hadamard products of theV λ with complex-valued harmonic mappings of the disk.

Jackson-Type Inequalities in the Space Sp

In the case of approximation of periodic functions in the space Sp, we determine the exact constants in Jackson-type inequalities for the Zygmund, Rogosinski, and de la Vallee Poussin linear summation methods.

Approximation of Classes of Analytic Functions by de la Vallée-Poussin Sums

For upper bounds of the deviations of de la Vallee-Poussin sums taken over classes of functions that admit analytic extensions to a fixed strip of the complex plane, we obtain asymptotic equalities. In certain cases, these equalities give a solution of the corresponding Kolmogorov–Nikol'skii problem.

Approximate Solution of the de la Vallée Poussin Problem for a Differential Equation of Neutral Type by the Projection-Iterative Method

We solve the de la Vallee Poussin problem for a functional-differential equation by the projection-iterative method. We construct an algorithm, establish conditions sufficient for the convergence of the method, and present a computational scheme.

The boundedness of the Cauchy singular integral operator in weighted Besov type spaces with uniform norms

The mapping properties of the Cauchy singular integral operator with constant coefficients are studied in couples of spaces equipped with weighted uniform norms. Recently weighted Besov type spaces got more and more interest in approximation theory and, in particular, in the numerical analysis of polynomial approximation methods for Cauchy singular integral equations on an interval. In a scale of pairs of weighted Besov spaces the authors state the boundedness and the invertibility of the Cauchy singular integral operator. Such result was not expected for a long time and it will affect further investigations essentially. The technique of the paper is based on properties of the de la Vallee Poussin operator constructed with respect to some Jacobi polynomials.

Marcinkiewicz- $$\theta$$ -Summability of Fourier Transforms

A general summability method of two-dimensional Fourier transforms is given with the help of an integrable function \(\theta\). Under some conditions on \(\theta\) we show that the maximal operator of the Marcinkiewicz-\(\theta\)-means of a tempered distribution is bounded from \(H_p \left( {R^2 } \right)\) to \(L_p \left( {R^2 } \right)\) for all \(p_0 < p \leqq \infty \) and, consequently, is of weak type \(\left( {1,1} \right)\), where \(p_0 < 1\) depends only on \(\theta \). As a consequence we obtain a generalization for Fourier transforms of a summability result due to Marcinkievicz and Zhizhiashvili, more exactly, the Marcinkiewicz-\(\theta \)-means of a function \(f \in L_1 \left( {R^2 } \right)\) converge a.e. to the function in question. Moreover, we prove that the Marcinkiewicz-\(\theta\)-means are uniformly bounded on the spaces \(H_p \left( {R^2 } \right)\) and so they converge in norm \(\left( {p_0 < p < \infty } \right)\). Some special cases of the Marcinkievicz-\(\theta \)-summation are considered, such as the Weierstrass, Picar, Bessel, Fejer, de la Vallee-Poussin, Rogosinski and Riesz summations.

Approximation Properties of the de la Vallée-Poussin Method

We present a survey of results concerning the approximation of classes of periodic functions by the de la Vallee-Poussin sums obtained by various authors in the 20th century.

Approximation of periodic functions by Vallee Poussin sums

Vallee-Poussin sums are introduced and their approximation properties for classes of continuous 2$\pi$-periodic functions are studied.

Approximation of $$\overline \psi$$ -Integrals of Periodic Functions by de la Vallée-Poussin Sums (Low Smoothness)

We investigate the asymptotic behavior of the upper bounds of deviations of linear means of Fourier series from the classes $$C_\infty ^{\psi}$$ . In particular, we obtain asymptotic equalities that give a solution of the Kolmogorov–Nikol'skii problem for the de la Vallee-Poussin sums on the classes $$C_\infty ^{\overline \psi }$$ .

A unified view of classical summation kernels

We treat kernels of the form % MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXanrfitLxBI9gBaerbd9wDYLwzYbItLDharqqt% ubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq% -Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0x% fr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuam% aaBaaaleaacaaIXaGaaGimaaqabaGccqGH9aqpciGGSbGaaiOBaiaa% ysW7caWGRbWaaSbaaSqaaiaadsfacaaIXaaabeaakiaac+cacaWGRb% WaaSbaaSqaaiaadsfacaaIYaaabeaakiabg2da9iabgkHiTmaabmaa% baGaamyramaaBaaaleaacaWGHbaabeaakiaac+cacaWGsbaacaGLOa% GaayzkaaGaey41aq7aaiWaaeaadaqadaqaaiaadsfadaWgaaWcbaGa% aGOmaaqabaGccqGHsislcaWGubWaaSbaaSqaaiaaigdaaeqaaaGcca% GLOaGaayzkaaGaai4laiaacIcacaWGubWaaSbaaSqaaiaaikdaaeqa% aOGaaGjbVlaadsfadaWgaaWcbaGaamysaaqabaGccaGGPaaacaGL7b% GaayzFaaaaaa!5C4A! $$k_n(t)={\mathop \sum^{n}\limits_{k=-n}g_n\ \big({k\over n}\big)\ e^{ikt}}$$ and give conditions on the functions gn such that the family (kn) is (or is not) a summation kernel. This applies to many classical kernels such as Dirichlet, Fejér, de la Vallée-Poussin, (C, α)-, Blackman and other summation kernels.

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 Vallee–Poussin problem and obtain the Pontryagin maximum principle in sufficient form.

Some Interpolating Operators of de la Vallée-Poussin Type

We consider discrete versions of the de la Vallee-Poussin algebraic operator. We give a simple sufficient condition in order that such discrete operators interpolate, and in particular we study the case of the Bernstein-Szegő weights. Furthermore we obtain good error estimates in the cases of the sup-norm and L1-norm, which are critical cases for the classical Lagrange interpolation.

On the Very Strong Summability of Orthogonal Series

It is well known that not every summability method implies the strong summability with any positive exponent. We give easygoing additional conditions on the terms of a positive regular Toeplitz-matrix implying the strong summability for any positive exponent. The classical (C, α > 0)- and Abel-summabilities satisfy our conditions plainly. We treat the generalized Abel, the Euler, the Riesz and the generalized de la Vallee Poussin methods, as well.

An extension of Bowen's dynamic voting rule to many dimensions

We ask in this paper about the effect on social decisions of limiting the size of changes that voters may propose each time in an otherwise standard dynamic social choice model. The voting rule we study can be seen as an extension of Bowen’s dynamic “majority voting” rule, and is closely related to the dynamic procedures for public good allocation in the literature (Dreze and de la Vallee Poussin 1971; Malinvaud 1971; Laffont and Maskin 1983; Chander 1993). Under general assumptions we prove existence and Pareto efficiency of equilibrium, and show that our rule motivates voters not to misrepresent preferences (more precisely, the rule is Strongly Locally Individually Incentive Compatible). Under Euclidean preferences we find that electoral cycles do not arise (i.e., the rule is convergent), that there is a unique equilibrium, and that the equilibrium coincides with the solution to an old problem of geometry, first addressed by Fermat, Torricelli, and Cavallieri.