Papers by Donatella Occorsio
217 Publications 1,939 Citations See Profile
Optimal systems of nodes for Lagrange interpolation on bounded intervals. A survey
Mathematics of Computation, Jul 1, 1990
The authors give a procedure to construct extended interpolation formulae and prove some uniform ... more The authors give a procedure to construct extended interpolation formulae and prove some uniform convergence theorems.
Dolomites Research Notes on Approximation, 2017
In the present paper the authors propose two numerical methods to approximate Hadamard transforms... more In the present paper the authors propose two numerical methods to approximate Hadamard transforms of the type H p (f w β , t) = = f (x) (x − t) p+1 w β (x)d x, where p is a nonnegative integer and w β (x) = e −|x| β , β > 1, is a Freud weight. One of the procedures employed here is based on a simple tool like the "truncated" Gaussian rule conveniently modified to remove numerical cancellation and overflow phenomena. The second approach is a process of simultaneous approximation of the functions {H k (f w β , t)} p k=0. This strategy can be useful in the numerical treatment of hypersingular integral equations. The methods are shown to be numerically stable and convergent and some error estimates in suitable Zygmund-type spaces are proved. Numerical tests confirming the theoretical estimates are given. Comparisons of our methods among them and with other ones available in literature are shown.
Special Issue on Functional Analysis, Approximation Theory and Numerical Analysis (FAATNA20>22)
Applied Numerical Mathematics
Constructive mathematical analysis, Jun 1, 2021
In this paper, some recent applications of the so-called Generalized Bernstein polynomials are co... more In this paper, some recent applications of the so-called Generalized Bernstein polynomials are collected. This polynomial sequence is constructed by means of the samples of a continuous function f on equispaced points of [0, 1] and depends on an additional parameter which can be suitable chosen in order to improve the rate of convergence to the function f , as the smoothness of f increases, overcoming the well-known low degree of approximation achieved by the classical Bernstein polynomials or by the piecewise polynomial approximation. The applications considered here deal with the numerical integration and the simultaneous approximation. Quadrature rules on equidistant nodes of [0, 1] are studied for the numerical computation of ordinary integrals in one or two dimensions, and usefully employed in Nyström methods for solving Fredholm integral equations. Moreover, the simultaneous approximation of the Hilbert transform and its derivative (the Hadamard transform) is illustrated. For all the applications, some numerical details are given in addition to the error estimates, and the proposed approximation methods have been implemented providing numerical tests which confirm the theoretical estimates. Some open problems are also introduced.
arXiv (Cornell University), Jan 12, 2021
The present paper concerns filtered de la Vallée Poussin (VP) interpolation at the Chebyshev node... more The present paper concerns filtered de la Vallée Poussin (VP) interpolation at the Chebyshev nodes of the four kinds. This approximation model is interesting for applications because it combines the advantages of the classical Lagrange polynomial approximation (interpolation and polynomial preserving) with the ones of filtered approximation (uniform boundedness of the Lebesgue constants and reduction of the Gibbs phenomenon). Here we focus on some additional features that are useful in the applications of filtered VP interpolation. In particular, we analyze the simultaneous approximation provided by the derivatives of the VP interpolation polynomials. Moreover, we state the uniform boundedness of VP approximation operators in some Sobolev and Hölder-Zygmund spaces where several integro-differential models are uniquely and stably solvable.
Mathematics of Computation, Sep 1, 1990
The authors give a procedure to construct extended interpolation formulae and prove some uniform ... more The authors give a procedure to construct extended interpolation formulae and prove some uniform convergence theorems.
arXiv (Cornell University), Aug 1, 2020
The paper deals with a special filtered approximation method, which originates interpolation poly... more The paper deals with a special filtered approximation method, which originates interpolation polynomials at Chebyshev zeros by using de la Vallée Poussin filters. These polynomials can be an useful device for many theoretical and applicative problems since they combine the advantages of the classical Lagrange interpolation, with the uniform convergence in spaces of locally continuous functions equipped with suitable, Jacobi-weighted, uniform norms. The uniform boundedness of the related Lebesgue constants, which equals to the uniform convergence and is missing from Lagrange interpolation, has been already proved in literature under different, but only sufficient, assumptions. Here, we state the necessary and sufficient conditions to get it. These conditions are easy to check since they are simple inequalities on the exponents of the Jacobi weight defining the norm. Moreover, they are necessary and sufficient to get filtered interpolating polynomials with a near best approximation error, which tends to zero as the number n of nodes tends to infinity. In addition, the convergence rate is comparable with the error of best polynomial approximation of degree n, hence the approximation order improves with the smoothness of the sought function. Several numerical experiments are given in order to test the theoretical results, to make a comparison with the Lagrange interpolation at the same nodes and to show how the Gibbs phenomenon can be strongly reduced.
Electronic Transactions on Numerical Analysis, 2023
Numerical Methods for Fredholm Integral Equations on the square
ABSTRACT In this paper we shall investigate the numerical solution of two-dimensional Fredholm in... more ABSTRACT In this paper we shall investigate the numerical solution of two-dimensional Fredholm integral equations by Nyström and collocation methods based on the zeros of Jacobi orthogonal polynomials. The convergence, stability and well conditioning of the method are proved in suitable weighted spaces of functions. Some numerical examples illustrate the efficiency of the methods.
Remote Sensing
Image resizing (IR) has a crucial role in remote sensing (RS), since an image’s level of detail d... more Image resizing (IR) has a crucial role in remote sensing (RS), since an image’s level of detail depends on the spatial resolution of the acquisition sensor; its design limitations; and other factors such as (a) the weather conditions, (b) the lighting, and (c) the distance between the satellite platform and the ground targets. In this paper, we assessed some recent IR methods for RS applications (RSAs) by proposing a useful open framework to study, develop, and compare them. The proposed framework could manage any kind of color image and was instantiated as a Matlab package made freely available on Github. Here, we employed it to perform extensive experiments across multiple public RS image datasets and two new datasets included in the framework to evaluate, qualitatively and quantitatively, the performance of each method in terms of image quality and statistical measures.
Journal of Mathematical Imaging and Vision
We present a new image scaling method both for downscaling and upscaling, running with any scale ... more We present a new image scaling method both for downscaling and upscaling, running with any scale factor or desired size. The resized image is achieved by sampling a bivariate polynomial which globally interpolates the data at the new scale. The method’s particularities lay in both the sampling model and the interpolation polynomial we use. Rather than classical uniform grids, we consider an unusual sampling system based on Chebyshev zeros of the first kind. Such optimal distribution of nodes permits to consider near-best interpolation polynomials defined by a filter of de la Vallée-Poussin type. The action ray of this filter provides an additional parameter that can be suitably regulated to improve the approximation. The method has been tested on a significant number of different image datasets. The results are evaluated in qualitative and quantitative terms and compared with other available competitive methods. The perceived quality of the resulting scaled images is such that impor...
Dolomites Research Notes on Approximation, 2016
In the present paper is proposed a numerical method to approximate Hilbert transforms of the type
arXiv (Cornell University), Aug 1, 2020
The paper deals with the approximate solution of integro-differential equations of Prandtl's type... more The paper deals with the approximate solution of integro-differential equations of Prandtl's type. Quadrature methods involving "optimal" Lagrange interpolation processes are proposed and conditions under which they are stable and convergent in suitable weighted spaces of continuous functions are proved. The efficiency of the method has been tested by some numerical experiments, some of them including comparisons with other numerical procedures. In particular, as an application, we have implemented the method for solving Prandtl's equation governing the circulation air flow along the contour of a plane wing profile, in the case of elliptic or rectangular wing-shape.
A method to approximate Hadamard finite part transforms on the positive semiaxis
The approximation of Hp(f) is of interest in different contexts. For instance, in the solution of... more The approximation of Hp(f) is of interest in different contexts. For instance, in the solution of hypersingular integral equations coming from Neumann 2D elliptic problems on a half-plane (see [1]). To our knowledge, most of the papers available in the literature deal with the approximation of Hadamard integrals on bounded intervals (see for instance [3], [4] and the references therein). In [1], the case of unbounded intervals is reduced to bounded ones by means of suitable transformations. Finally, in [2] we have proposed a method for approximating the integral in (1) for any fixed t by means of a suitable truncated Gauss-Laguerre rule. However, when “many” values of Hp(f, t) have to be computed, it can be more convenient the procedure we go to describe. Setting
Bit Numerical Mathematics, Apr 3, 2023
In this paper we consider the problem of the approximation of definite integrals on finite interv... more In this paper we consider the problem of the approximation of definite integrals on finite intervals for integrand functions showing some kind of "pathological" behavior, e.g. "nearly" singular functions, highly oscillating functions, weakly singular functions, etc. In particular, we introduce and study a product rule based on equally spaced nodes and on the constrained mock-Chebyshev least squares operator. Like other polynomial or rational approximation methods, this operator was recently introduced in order to defeat the Runge phenomenon that occurs when using polynomial interpolation on large sets of equally spaced points. Unlike methods based on piecewise approximation functions, mainly used in the case of equally spaced nodes, our product rule offers a high efficiency, with performances slightly lower than those of global methods based on orthogonal polynomials in the same spaces of functions. We study the convergence of the product rule and provide error estimates in subspaces of continuous functions. We test the effectiveness of the formula by means of several examples, which confirm the theoretical estimates.
arXiv (Cornell University), Jul 18, 2022
This paper provides a product integration rule for highly oscillating integrands, based on equall... more This paper provides a product integration rule for highly oscillating integrands, based on equally spaced nodes. The stability and the error estimate are proven in the space of continuous functions, and some numerical tests which confirm such estimates are provided.
Annali Dell'universita' Di Ferrara, Sep 9, 2022
In this paper we consider a numerical scheme for the treatment of an integro-differential equatio... more In this paper we consider a numerical scheme for the treatment of an integro-differential equation. The latter represents the formulation of a nonlocal diffusion type equation. The discretization procedure relies on the application of the line method. However, quadrature formulae are needed for the evaluation of the integral operator. They are based on generalized Bernstein polynomials. Numerical evidence shows that the proposed method is a suitable and reliable approach for the problem.
arXiv (Cornell University), Sep 22, 2021
A product quadrature rule, based on the filtered de la Vallée Poussin polynomial approximation, i... more A product quadrature rule, based on the filtered de la Vallée Poussin polynomial approximation, is proposed for evaluating the finite Hilbert transform in [−1, 1]. Convergence results are stated in weighted uniform norm for functions belonging to suitable Besov type subspaces. Several numerical tests are provided, also comparing the rule with other formulas known in literature.
Numerical Algorithms, Feb 5, 2021
In order to solve Prandtl-type equations we propose a collocation-quadrature method based on VP f... more In order to solve Prandtl-type equations we propose a collocation-quadrature method based on VP filtered interpolation at Chebyshev nodes. Uniform convergence and stability are proved in a couple of Hölder-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 L 2 case by cutting 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.
Uploads
Papers by Donatella Occorsio