An account of the error and of the convergence theory is given for Gauss-Laguerre quadrature formulae. We develop also truncated models of the original Gauss rules to compute integrals extended over the positive real axis.

We discuss solvability properties of a nonlinear hypersingular integral equation of Prandtl's type using monotonicity arguments together with different collocation iteration schemes for the numerical solution of such equations.

The paper gives a contribution of wavelet aspects to classical algebraic polynomial approximation theory. Algebraic polynomial interpolating scaling functions and wavelets are constructed by using the interpolating properties of de la Vallée Poussin kernels w.r.t. the four kinds of Chebyshev weights. For the decomposition and reconstruction of a given function the structure of the involved matrices is studied in order to reduce the computational effort by means of fast cosine and sine transforms.

A purely flexural mechanical analysis has been carried out for a thin, solid, circular plate deflected by a static transverse central force and bilaterally supported along two antipodal periphery arcs, the remaining part of the boundary being free. Monegato and Strozzi [6,7] have considered two particular contact reactions: the case where only a distributed force takes place, and the situation in which a distributed force is jointed to a distributed couple of properly selected profile. Both of these problems can been formulated in terms of an integral equation of the Prandtl type with Hilbert and Volterra operators, associated with two constraints conditions. Capobianco, Criscuolo and Junghanns [2] have studied an integro--differential equation of Prandtl type and a collocation method as well as a quadrature method for its approximate solution in weighted Sobolev spaces. Furthermore, collocation and collocation--quadrature methods for the same integral equation have been studied in weighted spaces of continuous functions \cite{CCJL}. The aim of the present paper is to present an algorithm related to the cited numerical model based on the collocation methods with quadrature methods on orthogonal polynomials as in \cite{CCJ,CCJL}. The optimal convergence rates presented here generalize the results shown in [7].

After some remarks on the convergence order of the classical gaussian formula for the numerical evaluation of integrals on unbounded interval, the authors develop a new quadrature rule for the approximation of such integrals of interest in the practical applications. The convergence of the proposed algorithm is considered and some numerical examples are given.

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 proved.

In this paper we analyze the numerical solution by a collocation method of a hypersingular integral equation resulting from the boundary value problem related to an infinite strip containing an edge crack perpendicular to its boundaries. Moreover, we show convergence results as well as numerical tests in a case of interest in fracture mechanics

In this paper we consider a collocation and a discrete collocation method for a Volterra integral equation with logarithmic perturbation kernel. We prove convergence and stability of these methods in a pair of Sobolev type spaces.

We deal with the numerical evaluation of the Hilbert transform on the real line by a Gauss type quadrature rule. The convergence and the stability of the method are investigated. The goodness of the numerical results for practical applications is examined.

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

The mathematical model of some environmental physics problems is represented by a singular integral equation with an oscillatory kernel. We investigate a method 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. In this way, we obtain a procedure that is numerically stable and the algorithm can be implemented in a fast way yielding satisfactory numerical results. Bounds of the error and of the amplification factor are also provided.

Using the de la Vallèe Poussin interpolation at the Chebyshev zeros, the authors construct polynomial interpolating wavelets and give the corresponding decomposition and reconstruction algorithms. The involved matrices can be diagonalized by sine and cosine orthogonal matrices. So the algorithms can be realized using fast sine and cosine transforms.

An integro-differential equation of Prandtl's type and a collocation method as well as a collocationquadrature method for its approximate solution is studied in weighted spaces of continuous functions.

We consider a Cauchy singular integral equation on the real line. A direct numerical mehod for solving this integral equation is given. We prove the convergence of the proposed method.

The numerical resolution of a Cauchy singular integral equation on the real line is stricly related with the good approximation of the Hilbert transform. In this paper we consider a numerical method besed on an approximation of the Hilbert transform and for this we prove the convergence in weighted uniform spaces.

We deal with the numerical evaluation of the Hilbert transform on the real line by a Gauss type quadrature rule. The convergence and the stability of the method are investigated.