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Recently, Mastroianni and Monegato derived error estimates for a numerical approach to evaluate the integral ∫ab ∫-11 f(x, y)/x-y dxdy, where (a,b) ≡ (-1,1) or (a,b) ≡ (a,-1) or (a,b) ≡ (1, b) and f(x,y) is a smooth function (see G. Mastroianni and G. Monegato, Error estimates in the numerical evaluation of some BEM singular integrals, Math. Comp. 70 2001, 251-267). The error bounds for the quadrature rule approximating the inner integral given in Theorems 3, 4 of that paper are not correct according to the proof. However, following a different approach, we are able to improve the pointwise error estimates given in that paper.

Interpolatory quadrature rules have been well studied and widely applied for numerical integration in the presence of simple singularities. However, these rules have been used less for strongly singular integrals due to the fact that divergence and instability problems increase with respect to the order of the singularity. In the present paper we study this aspect proving convergence and stability of suitable interpolatory rules to compute certain singular and hypersingular integrals. The particular case of these integrals for oscillatory functions is also examined. Some numerical tests confirming our theoretical analysis are also presented.

An algorithm for the approximate evaluation of the Hilbert transform has been proposed. The convergence of the procedure is proved. The stability of the algorithm is considered and some numerical examples are given.

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.

We examine the convergence of product quadrature formulas of interpolatory type, based on the zeros of certain generalized Jacobi polynomials, for the discretization of integrals of the type \[ ∫ − 1 1 K ( x , y ) f ( x ) d x , − 1 ≤ y ≤ 1 , \int _{ - 1}^1 {K(x,y)f(x)\,dx,} \quad - 1 \leq y \leq 1, \] where the kernel K ( x , y ) K(x,y) is weakly singular and the function f ( x ) f(x) has singularities only at the endpoints ± 1 \pm 1 . In particular, when K ( x , y ) = log ⁡ | x − y | K(x,y) = \log |x - y| , K ( x , y ) = | x − y | v K(x,y) = |x - y{|^v} , v > − 1 v > - 1 , and f ( x ) f(x) has algebraic singularities of the form ( 1 ± x ) σ {(1 \pm x)^\sigma } , σ > − 1 \sigma > - 1 , we prove that the uniform rate of convergence of the rules is O ( m − 2 − 2 σ log 2 m ) O({m^{ - 2 - 2\sigma }}{\log ^2}m) in the case of the first kernel, and O ( m − 2 − 2 σ − 2 v log ⁡ m ) O({m^{ - 2 - 2\sigma - 2v}}\log m) if v ≤ 0 v \leq 0 , or O ( m − 2 − 2 σ log ⁡ m ) O({m^{ - 2 - 2\sigma }}\log m) if v > 0 v > 0 , for the second, where m is the number of points in the quadrature rule.

The authors develop an algorithm for the numerical evaluation of Cauchy principal value integrals of oscillatory functions. The method is based on an interpolatory procedure at the zeros of the orthogonal polynomials with respect to a Jacobi weight. A numerically stable procedure is obtained and the corresponding algorithm can be implemented in a fast way yielding satisfactory numerical results. Bounds of the error and of the amplification factor are also provided.

An algorithm for the approximate evaluation of integrals defined by Cauchy principal value or by Hadamard finite part has been proposed. The convergence of the procedure is proved. The stability of the algorithm is considered and some numerical examples are given.

The authors study the error of the product quadrature rules to compute the integral\(\int_{ - 1}^1 {f\left( x \right) u\left( x \right)dx} \). Estimates inL 1-weighted norm are established whenu is a weight with algebraic and/or logarithmic singularities and the quadrature points are classical Jacobi zeros. Upper bounds for the generalized functions of second kind are also given.

Uniform convergence of Lagrange interpolation at the zeros of Jacobi polynomials in the presence of constraints is investigated. We show that by a simple procedure it is always possible to transform the matrices of these zeros into matrices such that the corresponding Lagrange interpolating polynomial respecting the given constraints well approximates a given function and its derivatives.

In numerical analysis it is important to construct interpolating polynomials approximating a given function and its derivatives simultaneously. The authors define some new good interpolating matrices with "many" nodes close to the endpoints of the interval and also give error estimates.

Weighted Lp convergence of derivatives of Hermite, interpolation on the zeros of Jacobi polynomials plus additional points is investigated.

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

Si dimostra un teorema di convergenza relativo a formule di quadratura di tipo gaussiano per il calcolo delle derivate di integrali a valor principale secondo Cauchy con peso. Si assegnano anche stime dell'errore dipendenti dalla regolarita della funzione «densita».

The authors consider product rules of Gauss type for the numerical approximation of certain two-dimensional Cauchy principal value integrals with respect to generalized smooth Jacobi weight functions. Convergence results for these rules are given, and some asymptotic estimates of the remainder are established.

In this paper we give some bound for generalized Jacobi functions of the second kind and their derivatives. Previous results on this subject are improved and generalized.

Product rules of interpolatory type on the zeros of generalized smooth Jacobi polynomials for the numerical approximation of Cauchy principal value integrals and their derivatives are considered. Convergence results for these rules are given and some estimates of the remainder are established in relation to the regularity of the “density” function. Read More: http://epubs.siam.org/doi/abs/10.1137/0725043

L’autore considera formule di quadratura del tipo Gauss-Lobatto per il calcolo di integrali a valor principale secondo Cauchy rispetto ad un peso generalizzato regolare di Jacobi. Vengono dimostrati teoremi di convergenza e stabilite alcune stime dell’errore. Si generalizzano e migliorano, inoltre, precedenti risultati. Abstract The author considers Gauss-Lobatto rules for evaluating of Cauchy singular integrals with respect to a generalized smooth Jacobi weight. Convergence theorems are given, and some estimates of the remainder are established. The previous results are generalized and improved.

Si costruiscono formule di quadratura gaussiane per il calcolo di integrali a valor principale secondo Cauchy e si dimostrano alcuni teoremi di convergenza.

In a previous paper the authors proposed a modified Gaussian rule * m (wf;t)to compute the integral (wf;t) in the Cauchy principal value sense associated with the weightw, and they proved the convergence in closed sets contained in the integration interval. The main purpose of the present work is to prove uniform convergence of the sequence { * m (wf;t)} on the whole integration interval and to give estimates for the remainder term. The same results are shown for particular subsequences of the Gaussian rules m (wf;t) for the evaluation of Cauchy principal value integrals. A result on the uniform convergence of the product rules is also discussed and an application to the numerical solution of singular integral equations is made.

The authors develop an algorithm for the numerical evaluation of the finite Hilbert transform, with respect to non-standard weight functions, by a product quadrature rule. In particular, this algorithm allows us to deal with the weight functions with algebraic and/or logarithmic singularities in the interval [-1, 1], by using the Chebyshev points as quadrature nodes. The practical application of the rule is shown to be straightforward and to yield satisfactory numerical results. Convergence theorems are also given, when the nodes are the zeros of certain classical Jacobi polynomials and the weight is defined as a generalized Ditzian-Totik weight.

Collocation and quadrature methods for singular integro-differential equations of Prandtl's type are studied in weighted Sobolev spaces. A fast algorithm basing on the quadrature method is proposed. Convergence results and error estimates are given.

The authors give convergence theorems for interpolatory product rules for evaluating Cauchy singular integrals and obtain asymptotic estimates of the remainder. Some results, previously established by other authors, are generalized and improved.