# Donatella OccorsioUniversità degli Studi della Basilicata | UniBas · Department of Mathematics, Computer Science and Economics

Donatella Occorsio

Mathematics

## About

77

Publications

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Introduction

## Publications

Publications (77)

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.

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 illust...

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 enlarged) size we aim to get, we approach the problem at a continuous scale where the underlying function representin...

A product quadrature rule, based on the filtered de la Vallée Poussin polynomial approximation, is proposed for evaluating the finite weighted 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 wi...

This paper provides a Nyström method for the numerical solution of Volterra integral equations whose kernels contain singularities of algebraic type. It is proved that the method is stable and convergent in suitable weighted spaces. An error estimate is also given as well as several numerical tests are presented.

In this paper, we propose a suitable combination of two different Nyström methods, both
using the zeros of the same sequence of Jacobi polynomials, in order to approximate the solution
of Fredholm integral equations on [−1, 1]. The proposed procedure is cheaper than the Nyström
scheme based on using only one of the described methods . Moreover, we...

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 Hölder-Zygmund spaces of locally continuous functions. With respect to classical methods based on Lagrange interpolation...

We present a new image scaling method both for downscaling and upscaling, running with any scale factor or desired size. It is based on the sampling of an approximating bivariate polynomial, which globally interpolates the data and is defined by a filter of de la Vall\'ee Poussin type whose action ray is suitable regulated to improve the approximat...

A product quadrature rule, based on the filtered de la Vall\'ee 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 oth...

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 g...

The paper deals with a special filtered approximation method, which originates interpolation polynomials at Chebyshev zeros by using de la Vallée Poussin filters. In order to get an optimal approximation in spaces of locally continuous functions equipped with weighted uniform norms, the related Lebesgue constants have to be uniformly bounded. In pr...

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...

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 fun...

The present paper concerns filtered de la Vall\'ee 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 (unif...

The paper deals with de la Vallée 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 conditio...

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 node...

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 test...

The paper deals with a special filtered approximation method, which originates interpolation polynomials at Chebyshev zeros by using de la Vall\'ee 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 conve...

In the present paper we introduce and study an extended product quadrature rule to approximate Hadamard finite part integrals of the type H p (f U, t) = = +∞ 0 f (x) (x − t) p+1 U(x)d x, t > 0, p ∈ , U(x) = e −x x γ , γ ≥ 0. Hypersingular integrals arise in many contexts, such as singular and hypersingular boundary integral equations, which are too...

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...

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...

The paper deals with the numerical approximation of integrals of the typeI(f,y):=∫−11f(x)k(x,y)dx,y∈S⊂R where f is a smooth function and the kernel k(x,y) involves some kinds of “pathologies” (for instance, weak singularities, high oscillations and/or endpoint algebraic singularities). We introduce and study a product integration rule obtained by i...

In the present paper we propose a product integration rule for hypersingular integrals on the positive semi-axis. The rule is based on an approximation of the density function f by a suitable truncated Lagrange polynomial. We discuss theoretical aspects by proving stability and convergence of the procedure for density functions f belonging to weigh...

The paper deals with the approximation of integrals of the type $$\begin{aligned} I(f;{\mathbf {t}})=\int _{{\mathrm {D}}} f({\mathbf {x}}) {\mathbf {K}}({\mathbf {x}},{\mathbf {t}}) {\mathbf {w}}({\mathbf {x}}) d{\mathbf {x}},\quad \quad {\mathbf {x}}=(x_1,x_2),\quad {\mathbf {t}}\in \mathrm {T}\subseteq \mathbb {R}^p, \ p\in \{1,2\} \end{aligned}...

In the present paper we consider hypersingular integrals of the following type (Formula presented) where the integral is understood in the Hadamard finite part sense, p is a positive integer, w α (x) = e −x x α is a Laguerre weight of parameter α ≥ 0 and t > 0. In [6] we proposed an efficient numerical algorithm for approximating (1), focusing our...

Let w(x)=e−xβxα, w¯(x)=xw(x) and let {pm(w)}m, {pm(w¯)}m be the corresponding sequences of orthonormal polynomials. Since the zeros of pm+1(w) interlace those of pm(w¯), it makes sense to construct an interpolation process essentially based on the zeros of Q2m+1:=pm+1(w)pm(w¯), which is called “Extended Lagrange Interpolation”. In this paper the co...

In this paper we propose a numerical procedure in order to approximate the solution of two-dimensional Fredholm integral equations on unbounded domains like strips, half-planes or the whole real plane. We consider global methods of Nyström types, which are based on the zeros of suitable orthogonal polynomials. One of the main interesting aspects of...

The authors propose a numerical method for computing Hilbert and Hadamard
transforms on (0;\infty) by a simultaneus approximation process involving a suitable
Lagrange polynomial of degree s and " truncated" Gaussian rule of order
m, with s<< m. The proposed procedure is convergent and pointwise error
estimates are given. Finally, some numerical te...

In this paper we propose some different strategies to approximate hypersingular integrals on the real semiaxis. Hadamard Finite Part integrals (shortly FP integrals), regarded as p th derivative of Cauchy principal value integrals, are of interest in the solution of hypersingular BIE, which model many different kind of Physical and Engineering prob...

In this paper we propose a global method to approximate the derivatives of the weighted Hilbert transform of a given function f Hp(fw?,t)=dpdtp?0+?f(x)x?tw?(x)dx=p!?0+?f(x)(x?t)p+1w?(x)dx, where p?{1,2,?}, t>0, and w?(x)=e?xx? is a Laguerre weight. The right-hand integral is defined as the finite part in the Hadamard sense. The proposed numerical a...

In the present paper the authors propose two numerical methods to approximate Hadamard transforms
on the real line. 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 approxima...

Let w ( x ) = e − x β x α , w ¯ ( x ) = x w ( x ) and denote by { p m ( w ) } m , { p n ( w ¯ ) } n the corresponding sequences of orthonormal polynomials. The zeros of the polynomial Q 2 m + 1 = p m + 1 ( w ) p m ( w ¯ ) are simple and are sufficiently far among them. Therefore it is possible to construct an interpolation process essentially based...

The paper introduces and studies the sequence of bivariate Generalized Bernstein operators $\{\mathbf{B}_{m,s}\}_{m,s},\quad m,s\in \mathbb{N},$
$$\mathbf{B}_{m,s}=I-(I-\mathbf{B}_{m})^s,\quad \mathbf{B}_{m}^i=\mathbf{B}_{m}(\mathbf{B}_{m}^{i-1}),$$
where $\mathbf{B}_{m}$ is the bivariate Bernstein operator.
These operators generalizes the ones int...

We study some extended Lagrange interpolation processes based
on the zeros of the generalized Laguerre polynomials. We give necessary and sufficient
conditions such that the convergence of these processes, in suitable Lp
weighted spaces on the real semiaxis, is assured for 1 < p < +∞.

Let $\{pm(w_\alpha)\}_m$ be the sequence of the polynomials orthonormal w.r.t. the Sonin–Markov
weight $w_\alpha(x) = e^{−x^2}|x|^\alpha$. The authors study extended Lagrange interpolation processes
essentially based on the zeros of $p_m(w_\alpha)p_{m+1}(w_\alpha), determining the conditions under
which the Lebesgue constants, in some weighted unif...

In this paper we investigate some Nystr¨om methods for Fredholm integral equations in the interval
[0, 1]. We give an overview of the order of convergence, which depends on the smoothness of the involved
functions. In particular, we consider the Nystr¨om methods based on the so called Generalized Bernstein
quadrature rule, on a Romberg scheme and o...

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...

In this brief survey are collected some recent results about optimal interpolation processes
of Lagrange type based on the zeros of generalized Laguerre polynomials, i.e. the sequence
of orthogonal polynomials {pm(w�)}m where w�(x) = e−x�
x�. A new extended Lagrange
process having optimal Lebesgue constants is also introduced

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 ef...

The author proposes a method to approximate the Hilbert transform on the real positive semiaxis by a suitable Lagrange interpolating polynomial. The method employs truncated Gaussian rules and uses the interlacing properties of the zeros of generalized Laguerre polynomials. The error estimate in a weighted uniform norm is proved and some numerical...

Let Bm(f) be the Bernstein polynomial of degree m. The generalized Bernstein polynomials B m,λ (f, x) = ∞ i=1 (−1) i+1 λ i B i m (f ; x), λ ∈ R + were introduced in [13]. In the present paper some of their properties are revisited and some applications are presented. Indeed, the stability and the convergence of a quadrature rule on equally spaced k...

Let w α (x)=e -x β x α , α>-1, β>1/2, be a generalized Laguerre weight, and denote by {p m (w α )} m the corresponding sequence of orthonormal polynomials. The starting point is that the polynomial Q 2m+1 =p m+1 (w α )p m (w α+1 ) has simple zeros and also well distributed in some sense. In view of this property, two different applications are desc...

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.

The author studies the uniform convergence of extended Lagrange interpolation processes based on the zeros of Generalized Laguerre polynomials.

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.

We study a truncated interpolation process based on the zeros of the Markov--Sonin polynomials and give convergence results
in some subspaces of the Lp weighted spaces.

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.

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.

This paper deals with the numerical approximation of a weakly singular integral transform by means of Laguerre nodes. Error estimates in a weighted uniform norm and some numerical tests are given.

In this brief survey special attention is paid to some recent procedures for constructing optimal interpolation processes, i.e., with Lebesgue constant having logarithmic behaviour. A new result on Lagrange interpolation based on the zeros of the associated Jacobi polynomials and on suitable additional nodes is given.

We introduce an interpolatory process essentially based on the Laguerre zeros and we prove that it is an optimal process in some weighted uniform spaces.

We discuss the approximation of integrals of type I(f ; t) = integral(R) f(x)K(x, t)e(-x2) vertical bar x vertical bar(2) dx, alpha > -1, where K is the weakly singular algebraic kernel vertical bar x - t vertical bar(lambda), -1 < lambda < 0, for "large" value of the parameter t. Moreover, we consider strongly oscillatory kernels of type K-1(x, t)...

The authors approximate a weakly singular integral operator on the real axis by a sequence of discrete operators based on Hermite zeros. The convergence and the stability of the resulting quadrature rule are stated with respect to the weighted uniform norm. Some numerical tests confirm the theoretical error estimates.

We consider product integration rules based on generalized Laguerre polynomials, for integrals of type I(f;t)=∫ 0 ∞ f(x)ψ(x,t)dx,t>0,f∈C LOC 0 [ 0 , ∞ ) where ψ(x,t) is a logarithmic or weakly singular algebraic kernel. In particular, we consider the cases of functions ψ(x,t) not necessarily positive and that of oscillating functions ψ(x,t)≡ψ(x). W...

Comparison between P´olya curves and generalized B´ezier (GB)
curves is performed with respect to the properties of interest for geometric
modelling of curvilinear forms. It turns out that the GB curve model shares
many important features with the P´olya model, and in some items, as stability,
fairness and closeness to a control polygon with large...

The distribution of the zeros of Jacobi polynomials of the second kind pm(wα, β) is studied, and lower and upper bounds for the corresponding Christoffel functions are given. Fourier expansions in the system {pm(wα, β)} are also considered. Moreover, some theorems on the convergence of Lagrange interpolating polynomials based on the zeros of pm(wα,...

Some algorithms are described for the numerical evaluation of Hadamard finite part integrals of type , where vα,β is a Jacobi weight. Convergence results and some numerical examples are given.

Application to CAGD of the generalized Bernstein operator Bm;k
introduced by G. Mastroianni and M.R.Occorsio [5] is discussed. The basic property of
this operator is that it generalizes both Bernstein and Lagrange operators. A
new free form curve scheme based on the Bm;k operator is introduced and some
properties of this scheme are discussed. Also...

In this paper, starting from interlacing properties of the zeros of the orthogonal polynomials, the authors propose a new method to approximate the finite Hilbert transform. For this method they give error estimates in uniform norm.

In this paper we study the convergence in weigthted Lp norm of an extended interpolation process on the zeros of a polynomial product of three orthogonal polynomials and on additional
knots.

An interpolation process on the zeros of a product of three orthogonal polynomials and on additional knots is studied.

The authors construct some extended interpolation formulae to approximate the derivatives of a function in uniform norm. They prove theorems on uniform convergence and give estimates of pointwise type and of simultaneous approximation.

The authors give a procedure to construct extended interpolation
formulae and prove some uniform convergence theorems.

## Projects

Projects (4)

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/

The Conference Functional Analysis, Approximation Theory and Numerical Analysis-FAATNA20>22 will be held in Matera, Italy, on July 5-8, 2022.
The goal of FAATNA20>22 is to bring together mathematicians who work in the fields of Functional Analysis, Approximation Theory and Numerical Analysis and to encourage collaboration and exchange of interdisciplinary ideas among the participants.
The conference topics include, but are not limited to:
- Approximation by linear or non linear operators
- Orthogonal polynomials
- Methods of functional analysis in approximation problems and/or differential equations
- Linear spaces and algebras of operators
- Function spaces
- Constructive approximation
- Numerical integration
- Numerical methods for integral and differential equations
- Numerical linear algebra.
The Plenary Speakers are
Prof. Francesco Altomare, University of Bari "Aldo Moro" (Italy)
Prof. José Bonet, Universitat Politècnica de València (Spain)
Prof. Len Bos, University of Verona (Italy)
Prof. Francisco Marcellán Español, Universidad Carlos III de Madrid (Spain)
Prof. Hrushikesh Mhaskar, Claremont Graduate University (USA)
Prof. Gradimir V. Milovanović, Serbian Academy of Sciences and Arts (Serbia)
Prof. Ioan Raşa, Technical University of Cluj-Napoca (Romania)
Prof. Gianluca Vinti, University of Perugia (Italy)
Prof. Yuesheng Xu, Old Dominion University (USA)
The scientific program is structured in Special Sessions which consist of invited and contributed talks. A Contributed Poster Session is also planned.
More details can be found at the website
http://web.unibas.it/faatna22/index.html