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Donatella Occorsio

Publications and source records attributed to Donatella Occorsio.

10 recordsLinked to original sources
arXiv2026

Lagrange interpolation processes based on the zeros of anti-Gauss Jacobi polynomials

This paper introduces and investigates a new Lagrange interpolation process based on the zeros of anti-Gauss Jacobi polynomials. Fundamental properties of anti-Gauss nodes, including their asymptotic distribution, are established, together with estimates for the associated polynomials and their derivatives. These results provide the basis for the construction of an interpolation process whose weighted Lebesgue constants exhibit logarithmic growth, ensuring optimal approximation properties. Compared with previously known interpolation schemes based on Jacobi nodes, the proposed process achieves optimal Lebesgue constants for a shifted range of endpoint weight parameters, allowing the use of smaller endpoint weight exponents. Convergence estimates are established for functions in suitable weighted Sobolev spaces, and numerical experiments support the theoretical findings.

math.NA↗
arXiv2026

A Rational Discrete Collocation Method for Second Kind Fredholm Equations

In this work we present a novel discrete collocation method for the numerical solution of Fredholm integral equations of the second kind in the space of continuous functions equipped with the uniform norm. The method is based on a rational interpolation scheme recently developed within the general framework of reproducing kernel Hilbert spaces. This rational approximation has no real poles, interpolates the target function at arbitrary Jacobi nodes and exhibits uniformly bounded Lebesgue constants. Moreover, it converges uniformly for all continuous functions at a rate at least equal to that of the best uniform polynomial approximation. These interesting properties are inherited by the resulting numerical method, for which stability, convergence and good conditioning are established under minimal assumptions on the integral kernel. A series of numerical experiments confirm the theoretical findings and indicate that, in the presence of particularly challenging kernels, the proposed approach provides a robust and effective alternative to Nyström-type methods.

math.NA↗
arXiv2026

De la Vallée Poussin type approximation for solving some Fredholm integral equations

In the present paper, we introduce a numerical method for second-kind Fredholm integral equations (FIEs) based on de la Vallée Poussin-type (VP) polynomial approximations at Jacobi zeros. This class of approximations offers several advantages over classical Lagrange interpolation at the same nodes. In particular, it guarantees uniformly bounded Lebesgue constants in suitable weighted function spaces and provides near-best uniform approximation for functions in these spaces, while also significantly mitigating the Gibbs phenomenon. We show how these properties can be exploited in the numerical solution of FIEs. In particular, the proposed approach effectively handles functions with possible algebraic endpoint singularities and kernel functions featuring weak singularities or highly oscillatory behavior. Under suitable assumptions, we prove stability and convergence of the method in weighted uniform spaces. Furthermore, we develop an efficient implementation based on the solution of a well-conditioned linear system. Numerical results confirm the theoretical error estimates and show that the proposed method achieves higher local accuracy than the corresponding Lagrange-based projection method.

math.NA↗
arXiv2022

On the numerical solution of Volterra integral equations on equispaced nodes

In the present paper, a Nystrom-type method for second kind Volterra integral equations is introduced and studied. The method makes use of generalized Bernstein polynomials, defined for continuous functions and based on equally spaced points. Stability and convergence are studied in the space of continuous functions, and some numerical tests illustrate the performance of the proposed approach.

math.NA↗
arXiv2022

Image scaling by de la Vallée-Poussin filtered interpolation

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 important details are preserved, and the appearance of artifacts is low. Competitive quality measurement values, good visual quality, limited computational effort, and moderate memory demand make the method suitable for real-world applications.

cs.CV↗
arXiv2021

Lagrange-Chebyshev Interpolation for image resizing

Image resizing is a basic tool in image processing and in literature we have many methods, based on different approaches, which are often specialized in only upscaling or downscaling. In this paper, independently of the (reduced or enhanced) size we aim to get, we approach the problem at a continuous scale where the underlying continuous image is globally approximated by the tensor product Lagrange polynomial interpolating at a suitable grid of first kind Chebyshev zeros. This is a well-known approximation tool that is widely used in many applicative yields, due to the optimal behavior of the related Lebesgue constants. Here we show how Lagrange-Chebyshev interpolation can be fruitfully applied also for resizing an arbitrary digital image in both downscaling and upscaling. The performance of the proposed method has been tested in terms of the standard SSIM and PSNR metrics. The results indicate that, in upscaling, it is almost comparable with the classical Bicubic resizing method with slightly better metrics, but in downscaling a much higher performance has been observed in comparison with Bicubic and other recent methods too. Moreover, in downscaling cases with an odd scale factor, we give an estimate of the mean squared error produced by our method and prove it is theoretically null (hence PSNR equals to infinite and SSIM equals to one) in absence of noise or initial artifacts on the input image.

math.NA↗
arXiv2020

Filtered interpolation for solving Prandtl's integro-differential equations

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 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 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.

math.NA↗
arXiv2020

On the filtered polynomial interpolation at Chebyshev nodes

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.

math.NA↗
arXiv2020

Quadrature methods for integro-differential equations of Prandtl's type in weighted spaces of continuous functions

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.

math.NA↗