Giuseppe Mastroianni

Giuseppe Mastroianni
Università degli Studi della Basilicata | UniBas · Department of Mathematics, Computer Science and Economics

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237
Publications
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Publications

Publications (237)
Article
The paper deals with weighted polynomial approximation for functions defined on (−1,1), which can grow exponentially both at −1 and at 1. We summarize recent results on function spaces with new moduli of smoothness, estimates for the best approximation, Lagrange interpolation, Fourier sums and Gaussian rules with respect to weights of the form w(x)...
Article
In order to approximate functions defined on the real semiaxis, which can grow exponentially both at $0$ and at $+\infty$, we introduce a suitable Lagrange operator based on the zeros of orthogonal polynomials with respect to the weight $w(x)=x^\gamma\mathrm{e}^{-x^{-\alpha}-x^\beta}$. We prove that this interpolation process has Lebesgue constan...
Article
In order to approximate functions defined on the real semiaxis, we introduce a new operator of Hermite–Fejér-type based on Laguerre zeros and prove its convergence in weighted uniform metric.
Article
We prove some results on the root-distances and the weighted Lebesgue function corresponding to orthogonal polynomials for Laguerre type exponential weights.
Chapter
The aim of this paper is to combine classical ideas for the theoretical investigation of the Nyström method for second kind Fredholm integral equations with recent results on polynomial approximation in weighted spaces of continuous functions on bounded and unbounded intervals, where also zeros of polynomials w.r.t. exponential weights are used.
Book
La teoria dell’approssimazione è un capitolo della Matematica presente in diversi contesti scientifici e in Analisi numerica costituisce la base teorica per la costruzione di varie procedure stabili e convergenti. Il volume, rivolto agli studenti della laurea magistrale in Matematica e agli studiosi della disciplina, tratta l’approssimazione trigon...
Article
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This paper deals with a special Hermite–Fejér interpolation process based at the zeros of generalized Freud polynomials which are orthogonal with respect to the weight w(x)=|x|αe-|x|β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackag...
Article
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This paper summarizes recent results on weighted polynomial approximations for functions defined on the real semiaxis. The function may grow exponentially both at 0 and at +∞. We discuss orthogonal polynomials, polynomial inequalities, function spaces with new moduli of smoothness, estimates for the best approximation, Gaussian rules, and Lagrange...
Article
Convergence-divergence properties of a barycentric operator are considered.
Article
A class of Fredholm integral equations of the second kind, with respect to the exponential weight function \(w(x)=\exp (-(x^{-\alpha }+x^\beta ))\), \(\alpha >0\), \(\beta >1\), on \((0,+\infty )\), is considered. The kernel k(x, y) and the function g(x) in such kind of equations, $$\begin{aligned} f(x)-\mu \int _0^{+\infty }k(x,y)f(y)w(y)\mathrm {...
Article
In order to approximate continuous functions on [0,+∞), we consider a Lagrange–Hermite polynomial, interpolating a finite section of the function at the zeros of some orthogonal polynomials and, with its first (r−1) derivatives, at the point 0. We give necessary and sufficient conditions on the weights for the uniform boundedness of the related ope...
Article
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We consider some 'truncated' Gaussian rules based on the zeros of the orthonormal polynomials w.r.t. The weight function with x ∈ (0, +∞), α >0 and β>1. We show that these formulas are stable and converge with the order of the best polynomial approximation in suitable function spaces. Moreover, we apply these results to the related Lagrange interpo...
Article
Two operators are introduced for the weighted L (p) approximation on the semiaxis with weight w (beta) (x) = x (alpha) exp(-x (beta) ), alpha > -1/p, 0 < beta a parts per thousand currency sign 1, allowing a much wider class of functions than in the case of classical Szasz-Mirakyan and Butzer operators. It is shown that in case 1/2 < beta a parts p...
Article
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In this paper the authors propose a Nystr\"om method based on a ``truncated" Gaussian rule to solve systems of Fredholm integral equations on the real line. They prove that it is stable and convergent and that the matrices of the solved linear systems are well-conditioned. Moreover, they give error estimates in weighted uniform norm and show some n...
Article
Full-text available
We consider the polynomial approximation on (0,+∞), with the weight u(x) = x^γ e^(−x^(−α)−x^β), α > 0, β > 1 and γ >= 0. We introduce new moduli of smoothness and related K-functionals for functions defined on the real semiaxis, which can grow exponentially both at 0 and at +∞. Then we prove the Jackson theorem, also in its weaker form, and the S...
Article
This paper is concerned with the stability of collocation methods for Cauchy singular integral equations with fixed singularities on the interval [-1, 1]. The operator in these equations is supposed to be of the form aL+bS+B-+/- with piecewise continuous functions a and b. The operator S is the Cauchy singular integral operator and B-+/- is a finit...
Article
Necessary and sufficient conditions for the weighted LpLp-convergence of Hermite and Hermite–Fejér interpolation of higher order based on Jacobi zeros are given, extending previous results for Lagrange interpolation. Error estimates in the weighted LpLp-norm are also shown.
Article
We state embedding theorems between spaces of functions defined on the real semi-axis, which can grow exponentially both at 0 and at +∞.
Poster
Full-text available
We state embedding theorems between function spaces related to spaces of functions defined on the real semiaxis, which can grow exponentially at 0 and +\infty.
Article
This paper is a recapitulation of the work of L. Szili and P. Vértesi [4] on multivariate Fourier series with triangular type partial sums. Namely, we give another proof for the corresponding lower estimation, which, in a way, is more direct than the previous one in [4].
Article
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We consider the weight u(x)=x^γ exp(−x^(−α)−x^β) , with x∈(0,+∞) , α > 0, β > 1 and γ ≥ 0 and prove Remez-, Bernstein–Markoff-, Schurand Nikolskii-type inequalities for algebraic polynomials with the weight u on (0, + ∞).
Article
The paper deals with the approximation of the solution of the following bivariate Fredholm integral equation f(y) - mu integral(D) K(x,y)f(x)(omega) over tilde (x)dx = g(y), y is an element of D, where the domain D is a triangle. The proposed procedure, by a suitable transformation, is essentially the Nystrom method based on the zeros of univariate...
Article
In order to approximate functions defined on (-1, 1) with exponential growth for vertical bar x vertical bar -> 1 we consider interpolation processes based on the zeros of orthonormal polynomials with respect to exponential weights. Convergence results and error estimates in weighted L-P metric and uniform metric are given. In particular, in some f...
Article
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In [1], we have introduced a new weighted type of modification of the classical Kantorovich operator. The advantage of this operator is that there is no restriction on the parameters of the weight, and the class of functions is wider than in the earlier version of the weighted operator (cf. the monograph of Ditzian and Totik [3]). Direct and conver...
Article
In some applications, one has to deal with the problem of integrating, over a bounded interval, a smooth function taking significant values, with respect to the machine precision or to the accuracy one wants to achieve, only in a very small part of the domain of integration. In this paper, we propose a simple and efficient numerical approach to com...
Article
This paper generalizes some results of L. B. Golinskii [4] on the asymptotic behaviour of reflection coefficients associated with generalized Jacobi weight functions.
Article
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We study the behavior of some “truncated” Gaussian rules based on the zeros of Pollaczek-type polynomials. These formulas are stable and converge with the order of the best polynomial approximation in suitable function spaces. Moreover, we apply these results to the related Lagrange interpolation process and to prove the stability and the convergen...
Article
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We consider a Lagrange–Hermite polynomial, interpolating a function at the Jacobi zeros and, with its first (r−1) derivatives, at the points ±1. We give necessary and sufficient conditions on the weights for the uniform boundedness of the related operator in certain suitable weighted Lp-spaces, 1 < p < 1, proving a Marcinkiewicz inequality involvin...
Article
Full-text available
We consider a Lagrange-Hermite polynomial, interpolating a function at the Jacobi zeros and, with its first (r - 1) derivatives, at the points +/- 1. We give necessary and sufficient conditions on the weights for the uniform boundedness of the related operator in certain suitable weighted L-p-spaces, 1 < p < infinity, proving a Marcinkiewicz inequa...
Article
We study the behavior of the Fourier sums in orthonormal polynomial systems, related to exponential weights on (−1,1), in weighted L1 and uniform metrics.
Article
We consider the weighted Hermite-Fejér interpolation process based on Jacobi nodes for classes of locally continuous functions defined by another Jacobi weight. Necessary and sufficient conditions for the weighted norm boundedness and for the convergence, as well as error estimates of the approximation, are given.
Article
Full-text available
The authors introduce a new procedure for the numerical treatment of Fredholm equations of the second kind on the real axis, based on a Nystrom method. The convergence of the method is proved and a priori estimates of the error are given. The case of kernels containing a Hilbert transform is also considered.
Article
In order to approximate functions defined on (−1,1) and having exponential singularities at the endpoints of the interval, we study the behavior of some modified Fourier Sums in an orthonormal system related to exponential weights. We give necessary and sufficient conditions for the boundedness of the related operators in suitable weighted Lp-space...
Article
We study the behavior of the Fourier sums in orthonormal polynomial systems, related to exponential weights on (−1,1), in weighted L1 and uniform metrics.
Article
Revisiting the results in [7], [8], we consider the polynomial approximation on (−1, 1) with the weight w(x)=exp(−(1−x2)^(−α)), α > 0. We introduce new moduli of smoothness, equivalent to suitable K-functionals, and we prove the Jackson theorem, also in its weaker form. Moreover, we state a new Bernstein inequality, which allows us to prove the Sal...
Article
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� In this papers the authors propose two projection methods to solve CSIE having smooth or weakly singular kernels. They prove their stability and convergence in Zygmund spaces equipped with uniform norm. Some numerical examples illustrating the accuracy of the methods are given.
Article
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In this paper the authors study ''truncated'' quadrature rules based on the zeros of Generalized Laguerre polynomials. Then, they prove the stability and the convergence of the introduced integration rules. Some numerical tests confirm the theoretical results.
Article
We state some pointwise estimates for the rate of weighted approximation of a continuous function on the semiaxis by polynomials. Similarly to a previous result in C[−1, 1] due to Z. Ditzian and D. Jiang [2], we consider weighted ϕ λ moduli of continuity, where 0 ≤ λ ≤ 1. The results we obtain bridge the gap between an old pointwise estimate by V.M...
Article
It is shown that for doubling weights, the zeros of the associated orthogonal polynomials are uniformly spaced in the sense that if cos θ m,k , θ m,k ∈[0,π] are the zeros of the m-th orthogonal polynomial associated with w, then θ m,k −θ m,k+1∼1/m. It is also shown that for doubling weights, neighboring Cotes numbers are of the same order. Finally,...
Article
Full-text available
The paper deals with the Lagrange interpolation of functions having a bounded variation derivative. For special systems of nodes, it is shown that this polynomial sequence converges with the best approximation order. The Lp weighted case is also discussed.
Article
Full-text available
In order to approximate functions defined on the real line or on the real semiaxis by polynomials, we introduce some new Fourier-type operators, connected to the Fourier sums of generalized Freud or Laguerre orthonormal sys- tems. We prove necessary and su±cient conditions for the boundedness of these operators in suitable weighted L^p-spaces, with...
Article
Necessary and sufficient conditions for the stability of certain collocation methods applied to Cauchy singular integral equations on an interval are presented for weighted L' norms. Moreover, the behavior of the approximation numbers, in particular their so-called k -splitting property, of the respective sequence of discretized operators is discus...
Article
Full-text available
We introduce an interpolation process based on some of the zeros of the mth generalized Freud polynomial. Convergence results and error estimates are given. In particular we show that, in some important function spaces, the interpolating polynomial behaves like the best approximation. Moreover the stability and the convergence of some quadrature ru...
Article
Numerical procedures to solve Fredholm integral equations of the second kind on the real semiaxis are proposed. Their stability and convergence are proved and error estimates in L p weighted norm are given. Numerical examples are also included.
Article
Full-text available
This survey is devoted to the study of the behaviour of the Fourier sums in weighted spaces of functions. The cases of functions supported on bounded or unbounded intervals of the real line are investigated. In particular the proof of the Riesz theorem is extended to the case of Fourier sums in systems of suitable orthonormal polynomials.
Article
Full-text available
In this paper we introduce some numerical methods for solving Fredholm integral equations of the second kind on the real semiaxis and prove that the proposed procedures are stable and convergent. Error estimates and numerical tests are also included.
Article
We show how truncated Gauss-Laguerre quadrature formulas can be used to produce accurate approximations and high rates of convergence, also when they are applied to integrand functions having only an algebraic type decay to zero at infinity. The approach presented in the paper is proposed for the computation of integrals and for the construction of...
Article
Bernstein polynomials are a useful tool for approximating functions. In this paper, we extend the applicability of this operator to a certain class of locally continuous functions. To do so, we consider the Pollaczek weight w(x)≔exp(−1x(1−x)),0x1, which is rapidly decaying at the endpoints of the interval considered. In order to establish convergen...
Article
In the cases A=[−1, 1] and A=[0,+∞), with , respectively, in this paper we consider the corresponding Fredholm integral equations of the second kind in the spaces of continuous functions equipped with certain uniform weighted norms. Assuming the continuity of the kernel k(x,y) we use Nyström methods and prove the stability, the convergence and the...
Article
We give error estimates for the weighted approximation of functions on the real line with Freud-type weights, by entire functions interpolating at finitely or infinitely many points on the real line. The error estimates involve weighted moduli of continuity corresponding to general Freud-type weights for which the density of polynomials is not alwa...
Article
Full-text available
In this paper, the authors introduce a Nyström method for solving systems of Fredholm integral equations on the real semiaxis. They prove that the method is stable and convergent. Moreover, they show some numerical tests that confirm the error estimates. Finally, they discuss a specific application to an inverse scattering problem for the Schröding...
Article
The authors investigate two sets of nodes for bivariate Lagrange interpolation in [-1, 1]2 and prove that the order of the corresponding Lebesgue constants is (log n)2.
Article
We show how truncated Gauss-Laguerre quadrature formulas can be used to produce accurate approximations and high rates of convergence, also when they are applied to integrand functions having only an algebraic type decay to zero at infinity. The approach presented in the paper is proposed for the computation of integrals and for the construction of...
Chapter
Full-text available
The second chapter is devoted to orthogonal polynomials on the real line and weighted polynomial approximation. In Sect. 2.1 we introduce the concept of orthogonality and consider several examples of orthogonal systems. In particular, we study Fourier expansions and best approximation. In Sect. 2.2 we give the basic properties of polynomials orthog...
Article
We prove that the weighted error of approximation by the Szász-Mirakyan-type operator introduced in [1] is equivalent to the modulus of smoothness of the function. This result is analogous to previous results for Bernstein-type operators obtained by Ditzian-Ivanov and Szabados.
Article
In order to approximate functions defined on (0, +∞), the authors consider suitable Lagrange polynomials and show their convergence in weighted L p -spaces.
Article
Full-text available
In this paper we consider the approximation of functions by suitable "truncated" Fourier Sums in the generalized Freud and Laguerre systems. We prove necessary and sufficient conditions for the uniform boundedness in Lp weighted spaces.
Chapter
Trigonometric approximation is considered in this chapter. We start this chapter with the Fourier operator and Fourier sums and give approximations by sums of Fourier and Fejér and de la Vallée Poussin means. Their discrete versions and the Lagrange trigonometric operator are also investigated. As a basic tool for studying approximating properties...
Chapter
Chapter 5 provides some selected applications in numerical analysis. In the first section on quadrature formulae we present some special Newton-Cotes rules, the Gauss-Christoffel, Gauss-Radau and Gauss-Lobatto quadratures, the so-called product integration rules, as well as a method for the numerical integration of periodic functions on the real li...
Chapter
Chapter 4 treats algebraic interpolation processes in the uniform norm, starting with the so-called optimal system of nodes , which provides Lebesgue constants of order log n and the convergence of the corresponding interpolation processes. Moreover, the error of such an approximation is near to the error of the best uniform approximation. Beside t...
Chapter
The first chapter provides an account of basic facts on approximation by algebraic and trigonometric polynomials introducing the most important concepts on approximation of functions. In Sect. 1.1 we introduce the basic notions, a connection between algebraic and trigonometric polynomials, best approximation by polynomials and give an account on Ch...
Article
Full-text available
We present a complete collection of results dealing with the polynomial approximation of functions on (0,+1).
Article
We consider polynomial approximation problems on the real line with generalized Freud weights. The generalization means multiplying these weights by so-called generalized polynomials which have finitely many roots on the corresponding intervals. Analogues of classical polynomial inequalities, as well as direct and converse approximation theorems, w...
Article
Full-text available
We study mapping properties, boundedness and invertibility of some Cauchy singular integral operators in a scale of pairs of Besov type subspaces of C(−1, 1). Our results include all those already available in the literature. Mathematics Subject Classification (2000). Primary 44A15; Secondary 41A10.
Article
We give error estimates for the weighted approximation of functions with singularities at the endpoints on the semiaxis by some modifications of Sz\'asz--Mirakyan operators. To do so, we define a new weighted modulus of smoothness and prove its equivalence to the weighted K-functional. Also, the class of functions for which the modified Sz\'asz--Mi...
Chapter
Some convergent and stable numerical procedures for Cauchy singular integral equations are given. The proposed approach consists of solving the regularized equation and is based on the weighted polynomial interpolation. The convergence estimates are sharp and the obtained linear systems are well conditioned.
Article
The authors describe the major results they have recently obtained on the truncation of interpolation processes on unbounded intervals, and on their applications to the numerical evaluation of corresponding integrals and to the resolution of a class of integral equations.
Article
Full-text available
In this paper the authors propose a numerical method for the approximate solution of some classes of Fredholm and Cauchy integral equations including the “discrete collocation” and “collocation” methods.
Chapter
In order to approximate functions on unbounded intervals, the authors show the convergence of truncated Fourier Sums and truncated Lagrange Polynomials.
Article
The authors consider the generalized airfoil equation in some weighted Hölder–Zygmund spaces with uniform norms. Using a projection method based on the de la Vallée Poussin interpolation, they find an approximate polynomial solution which converges to the original solution like the best uniform weighted polynomial approximation. The proposed numeri...
Article
The paper deals with the approximation of functions f on (0,+8), where f can be singular at the origin, by means of Bernstein-type sequences. Error estimates in weighted uniform spaces with some converse results are given.
Article
Full-text available
The authors obtain precise estimations for the coefficients of Hermite-Fejér interpolation of higher order based on Generalized Jacobi zeros.
Article
The paper deals with the approximation of integrals in R fw�, where wis a Markov–Sonin weight and f can be singular at the origin. Gaussian-type quadrature rules, having a better behaviour w.r.t. the ordinary Gaussian rule, are introduced.Error estimates in weighted L1 norm and some numerical tests are given.
Article
The authors give error estimates, a Voronovskaya-type relation, strong converse results and saturation for the weighted approximation of functions on the real line with Freud weights by Bernstein-type operators.
Article
Full-text available
The authors give direct and converse results for the weighted approximation of functions with endpoint or inner singularities by Bernstein operators. Key words: Weights with endpoint or inner zeros in [0, 1], Weighted modulus of smoothness and K-functional, Bernstein operator.
Conference Paper
The authors propose a simple numerical method to approximate the solution of CSIE.The convergence and the stability of the procedure are proved and some numerical examples are shown.
Article
We propose replacing the classical Gauss-Laguerre quadrature formula by a truncated version of it, obtained by ignoring the last part of its nodes. This has the effect of obtaining optimal orders of convergence. Corresponding quadrature rules with kernels are then considered and optimal error estimates are derived also for them. These rules are fin...
Article
The authors propose some numerical methods to solve Fredholm integral equations of the second kind on unbounded intervals. The proposed procedure includes projection methods and their discretized versions. Special attention is turned to the conditioning of the linear system corresponding to the finite-dimensional equation.
Article
A quadrature rule as simple as the classical Gauss formula, with a lower computational cost and having the same convergence order of best weighted polynomial approximation in L1 is constructed to approximate integrals on unbounded intervals. An analogous problem is discussed in the case of Lagrange interpolation in weighted L2 norm. The order of co...
Article
The authors state a one-sided convergence theorem for Lagrange interpolation based on certain generalizations of Jacobi weights. In a sense the results are best possible.
Article
 We consider a polynomial collocation for the numerical solution of a second kind integral equation with an integral kernel of Mellin convolution type. Using a stability result by Junghanns and one of the authors, we prove that the error of the approximate solution is less than a logarithmic factor times the best approximation and, using the asympt...
Article
We study the weighted uniform approximation of functions with inner singularities by exponential-type weights in [−1, 1]. Convergence theorems and pointwise and uniform approximation error estimates not possible by polynomials are proved.
Article
We propose replacing the classical Gauss--Laguerre quadrature formula by a truncated version of it, obtained by ignoring the last part of its nodes. This has the effect of obtaining optimal orders of convergence. Corresponding quadrature rules with kernels are then considered and optimal error estimates are derived also for them. These rules are fi...
Article
Full-text available
The authors propose some simple algorithms for computing the Cauchy principal value integrals on the real line, using zeros of Markov-Sonin polynomials. Error estimates are proved and some numerical tests are shown.
Article
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The authors estimate the error of best polynomial approximation in Lp-weighted spaces with weights having zeros inside [−1, 1], using a suit- able modulus of smoothness. The Jackson and Stechkin inequalities are given. Moreover some estimates of the derivatives of the polynomials of best approx- imation are proved.
Article
The authors construct interpolatory operators for the weighted approximation of functions with endpoints or inner singularities by iterated exponential-type weights in [-1,1]. Convergence theorems and uniform approximation error estimates not possible by polynomials are proved.
Article
We construct Shepard-type operators for weighted uniform approximation of functions with endpoint singularities by exponential-type weights in [−1,1]. Convergence theorems and pointwise and uniform approximation error estimates not possible by polynomials are proved.
Article
Full-text available
Let $$W: = \exp \left( { - Q} \right)$$ , where $$Q$$ is of smooth polynomial growth at $$\infty$$ , for example $$Q\left( x \right) = \left| x \right|^\beta ,\beta >1$$ . We call $$W^2 $$ a Freud weight. Let $$\left\{ {x_{j{\kern 1pt} n} } \right\}_{j = 1}^n $$ and $$\left\{ {\lambda _{j{\kern 1pt} n} } \right\}_{j = 1}^n $$ denote respecti...

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The "Research ITalian network on Approximation (RITA)" groups several italian numerical analysts working on Multivariate Approximation. Detailed Information about all research topics and activities of RITA members can be found at the website https://sites.google.com/site/italianapproximationnetwork/