In order to solve Prandtl--type equations we propose a collocation--quadrature method based on de la Vallée Poussin (briefly VP) filtered interpolation at Chebyshev nodes. Uniform convergence and stability are proved in a couple of Holder--Zygmund spaces of locally continuous functions. With respect to classical methods based on Lagrange interpolation at the same collocation nodes, we succeed in reproducing the optimal convergence rates of the L2 case and cut off the typical log factor which seemed inevitable dealing with uniform norms. Such an improvement does not require a greater computational effort. In particular, we propose a fast algorithm based on the solution of a simple 2-bandwidth linear system and prove that, as its dimension tends to infinity, the sequence of the condition numbers (in any natural matrix norm) tends to a finite limit.

The paper concerns the weighted uniform approximation of a real function on the d-cube [-1, 1]^d, with d > 1, by means of some multivariate filtered polynomials. These polynomials have been deduced, via tensor product, from certain de la Vallée Poussin type means on [-1, 1], which generalize classical delayed arithmetic means of Fourier partial sums. They are based on arbitrary sequences of filter coefficients, not necessarily connected with a smooth filter function. Moreover, in the continuous case, they involve Jacobi-Fourier coefficients of the function, while in the discrete approximation, they use function's values at a grid of Jacobi zeros. In both the cases we state simple sufficient conditions on the filter coefficients and the underlying Jacobi weights, in order to get near-best approximation polynomials, having uniformly bounded Lebesgue constants in suitable spaces of locally continuous functions equipped with weighted uniform norm. The results can be useful in the construction of projection methods for solving Fredholm integral equations, whose solutions present singularities on the boundary. Some numerical experiments on the behavior of the Lebesgue constants and some trials on the attenuation of the Gibbs phenomenon are also shown.

Si mostrano le condizioni sufficienti per avere costanti di Lebesgue ottimali (anche pesate) per l'approssimazione polinomiale discreta di una funzione di due variabili, nota su una griglia di zeri di Jacobi. Si considera sia l'interpolazione bivariata di Lagrange che l'approssimazione generalizzata di tipo de la Vallée Poussin, ottenuta mediante una generica funzione filtro. **

The paper deals with de la Vallee Poussin type interpolation on the square at tensor product Chebyshev zeros of the first kind. The approximation is studied in the space of locally continuous functions with possible algebraic singularities on the boundary, equipped with weighted uniform norms. In particular, simple necessary and sufficient conditions are proved for the uniform boundedness of the related Lebesgue constants. Error estimates in some Sobolev-type spaces are also given. Pros and cons of such a kind of filtered interpolation are analyzed in comparison with the Lagrange polynomials interpolating at the same Chebyshev grid or at the equal number of Padua nodes. The advantages in reducing the Gibbs phenomenon are shown by means of some numerical experiments. (C) 2020 Elsevier Inc. All rights reserved.

In the present paper, we propose a numerical method for the simultaneous approximation of the finite Hilbert and Hadamard transforms of a given function f, supposing to know only the samples of f at equidistant points. As reference interval we consider [-1,1] and as approximation tool we use iterated Boolean sums of Bernstein polynomials, also known as generalized Bernstein polynomials. Pointwise estimates of the errors are proved, and some numerical tests are given to show the performance of the procedures and the theoretical results.

Si considera l'interpolazione di una funzione di due variabili su una griglia di nodi di Chebyshev di I specie mediante polinomi di approssimazione filtrata basati sul classico filtro di de la Vallée Poussin. Tale problema trova applicazioni sia nell'analisi delle immagini che nella risoluzione numerica di equazioni integrali singolari. Vengono mostrate stime dell'errore in norma uniforme pesata che dipendono dai diversi gradi di regolarità della funzione approssimante. Inoltre confronti con l'interpolazione di Lagrange sugli stessi nodi e con l'interpolazione sullo stesso numero di Padua points, mostrano i vantaggi dell'interpolazione filtrata di de la Vallée Poussin in particolare nella riduzione del fenomeno di Gibbs. xx

The paper concerns the uniform polynomial approximation of a function f, continuous on the unit Euclidean sphere of $R^3$ and known only at a finite number of points that are somehow uniformly distributed on the sphere. First, we focus on least squares polynomial approximation and prove that the related Lebesgue constants w.r.t. the uniform norm grow at the optimal rate. Then, we consider delayed arithmetic means of least squares polynomials whose degrees vary from n - m up to n + m, being m = ?n for any fixed parameter 0 < ? < 1. As n tends to infinity, we prove that these polynomials uniformly converge to f at the near-best polynomial approximation rate. Moreover, for fixed n, by using the same data points, we can further improve the approximation by suitably modulating the action ray m determined by the parameter ?. Some numerical experiments are given to illustrate the theoretical results.

We consider the generalization of discrete de la Vallée Poussin means on the square, obtained via tensor product by the univariate case. Pros and cons of such a kind of filtered approximation are discussed. In particular, under simple, we get near-best discrete approximation polynomials in the space of all locally continuous functions on the square with possible algebraic singularities on the boundary, equipped with the weighted uniform norm. In the four Chebychev cases, these polynomials also interpolate the function. Moreover, for almost everywhere smooth functions, the Gibbs phenomenon appears reduced. Comparison with other interpolating polynomials are proposed.

We consider some discrete approximation polynomials, namely discrete de la Vallée Poussin means, which have been recently deduced from certain delayed arithmetic means of the Fourier-Jacobi partial sums, in order to get a near-best approximation in suitable spaces of continuous functions equipped with the weighted uniform norm. By the present paper we aim to analyze the behavior of such discrete de la Vallée means in weighted L1 spaces, where we provide error bounds for several classes of functions, included functions of bounded variation. In all the cases, under simple conditions on the involved Jacobi weights, we get the best approximation order. During our investigations, a weighted L1 Marcinkiewicz type inequality has been also stated.

In this paper, a general approach to de la Vallée Poussin means is given and the resulting near best polynomial approximation is stated by developing simple sufficient conditions to guarantee that the Lebesgue constants are uniformly bounded. Not only the continuous case but also the discrete approximation is investigated and a pointwise estimate of the generalized de Vallée Poussin kernel has been stated to this purpose. The theory is illustrated by several numerical experiments.

We estimate the error of Gauss-Jacobi quadrature rule applied to a function f, which is supposed locally absolutely continuous in some Besov type spaces, or of bounded variation on [-1,1]. In the first case the error bound concerns the weighted main part phi-modulus of smoothness of f introduced by Z. Ditzian and V. Totik, while in the second case we deal with a Stieltjes integral with respect to f. The stated estimates generalize several error bounds from literature and, in both the cases, they assure the same convergence rate of the error of best polynomial approximation in weighted L-1 space. (C) 2017 IMACS. Published by Elsevier B.V. All rights reserved.

We study a nonlinear boundary value problem involving a nonlocal (integral) operator in the coefficients of the unknown function. Provided sufficient conditions for the existence and uniqueness of the solution, for its approximation, we propose a numerical method consisting of a classical discretization of the problem and an algorithm to solve the resulting nonlocal and nonlinear algebraic system by means of some iterative procedures. The second order of convergence is assured by different sufficient conditions, which can be alternatively used in dependence on the given data. The theoretical results are confirmed by several numerical tests. (C) 2015 Elsevier B.V. All rights reserved.

The modeling of various physical questions often leads to nonlinear boundary value problems involving a nonlocal operator, which depends on the unknown function in the entire domain, rather than at a single point. In order to answer an open question posed by J.R. Cannon and D.J. Galiffa, we study the numerical solution of a special class of nonlocal nonlinear boundary value problems, which involve the integral of the unknown solution over the integration domain. Starting from Cannon and Galiffa's results, we provide other sufficient conditions for the unique solvability and a more general convergence theorem. Moreover, we suggest different iterative procedures to handle the nonlocal nonlinearity of the discrete problem and show their performances by some numerical tests.

In a recent paper we proposed a numerical method to solve a non-standard non-linear second order integro-differential boundary value problem. Here, we answer two questions remained open: we state the order of convergence of this method and provide some sufficient conditions for the uniqueness of the solution both of the discrete and the continuous problem. Finally, we compare the performances of the method for different choices of the iteration procedure to solve the non-standard nonlinearity.

Gaussian quadrature has been extensively studied in literature and several error estimates have been proved under dierent smoothness assumptions of the integrand function. In this talk we are going to state a general error estimate for Gauss-Jacobi quadrature, based on the weighted moduli of smoothness introduced by Z. Ditzian and V. Totik in [1]. Such estimate improves a previous result in [1, Theorem 7.4.1] and it includes several error bounds from literature as particular cases. Its proof has been achieved by using certain delayed means of the Fourier projections (de la Vallee Poussin means), which approximation properties will be also discussed. References [1] Z.Ditzian, V.Totik, Moduli of smoothness, SCMG Springer{Verlag, New York, 1987.

One of the most popular discrete approximating polynomials is the Lagrange interpolation polynomial and the Jacobi zeros provide a particularly convenient choice of the interpolation knots on [?1, 1]. However, it is well known that there is no point system such that the associate sequence of Lagrange polynomials, interpolating an arbitrary function f, would converge to f w.r.t. any weighted uniform or L1 norm. To overcome this problem, some discrete approximating polynomials have been originated from certain delayed arithmetic means of the Fourier-Jacobi partial sums (de la Vallee Poussin means) by approximating the Fourier coefficients with a Gaussian quadrature rule. The uniform convergence of these polynomials in suitable spaces of continuous functions has been recently proved. In this talk we complete the study by analyzing the approximation error w.r.t. the weighted L1 norm. In the main estimate we state, we use Ditzian-Totik moduli of smoothness.

The paper deals with the numerical solution of a nonstandard Sturm-Liouville boundary value problem on the half line where the coefficients of the differential terms depend on the unknown function by means of a scalar integral operator. By using a finite difference discretization, a truncated quadrature rule and an iterative procedure, we construct a numerical method, whose convergence is proved. The order of convergence and the truncation at infinity are also discussed. Finally, some numerical tests are given to show the performance of the method. © 2013 Elsevier B.V. All rights reserved.

The paper examines a particular class of nonlinear integro-differential equations consisting of a Sturm-Liouville boundary value problem on the half-line, where the coefficient of the differential term depends on the unknown function by means of a scalar integral operator. In order to handle the nonlinearity of the problem, we consider a fixed point iteration procedure, which is based on considering a sequence of classical Sturm-Liouville boundary value problems in the weak solution sense. The existence of a solution and the global convergence of the fixed-point iterations are stated without resorting to the Banach fixed point theorem. Moreover, the unique solvability of the problem is discussed and several examples with unique and non-unique solutions are given.

We consider a particular second-order integrodifferential boundary value problem arising from the kinetic theory of dusty plasmas, and we provide information on the existence and other qualitative properties of the solution that have been essential in the numerical investigation.