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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 yields the remarkable property of improving the rate of convergence to the function f, according with the smoothness of f. This means that the sequence does not suffer of the saturation phenomena occurring by using the classical Bernstein polynomials or arising in 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.

In this paper we propose a suitable combination of two collocation methods based on the zeros of Jacobi polynomials in order to approximate the solution of Fredholm integral equations on [−1,1]. One of the main interesting aspects of this procedure is that our approach is cheaper than the usual collocation method based on standard Lagrange interpolation using Jacobi zeros. Moreover, we can successfully manage functions with algebraic singularities at the endpoints. The error of the method is comparable with the error of the best polynomial approximation in suitable spaces of functions, equipped with the weighted uniform norm. The convergence and the stability of the method is proved, and some numerical tests, which confirm the theoretical estimates, are provided.

In this paper, the generalized Love integral equation has been considered. In order to approximate the solution, a Nyström method based on a mixed quadrature rule has been proposed. Such a rule is a combination of a product and a “dilation” quadrature formula. The stability and convergence of the described numerical procedure have been discussed in suitable weighted spaces and the efficiency of the method is shown by some numerical tests.

The paper deals with the numerical approximation of integrals of the type I ( f , y ) : = ∫ − 1 1 f ( x ) k ( x , y ) d x , 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 interpolating f by an extended Lagrange polynomial based on Jacobi zeros. We prove that the rule is stable and convergent with the order of the best polynomial approximation of f in suitable function spaces. Moreover, we derive a general recurrence relation for the new modified moments appearing in the coefficients of the rule, just using the knowledge of the usual modified moments. The new quadrature sequence, suitable combined with the ordinary product rule, allows to obtain a “mixed” quadrature scheme, significantly reducing the number of involved samples of f. Numerical examples are provided in order to support the theoretical results and to show the efficiency of the procedure.

In narrow open channels, the three-dimensional nature of the flow and the transport momentum from the sidewalls to the central region cause the maximum longitudinal velocity to occur below the water surface. The entropy model is unable to accurately describe the velocities near the free surface when the dip phenomenon exists. The present paper proposes a new dip-modified entropy law for steady open channel flows, which consists of three additional terms: the first one similar to Coles&rsquo, function, the second one linearly proportional to the logarithmic distance from the free surface, and the third one depending on the cubic correction near the maximum velocity. The validity of the new model was tested on a set of laboratory measurements carried out in a straight rectangular flume with smooth boundaries and for different values of water discharge, bottom slope, and aspect ratio. A detailed error analysis showed good agreement with the data measured through the present research and a more accurate prediction of the velocity-dip-position compared with the one evaluated through the original entropy model. In addition, the modified entropy wake law matched very well with other literature data collected in rectangular cross-sections with different flow conditions.

In this paper we investigate the numerical solution of Cauchy bisingular integral equations of the first kind on the square. We propose two different methods based on a global polynomial approximation of the unknown solution. The first one is a discrete collocation method applied to the original equation and then is a “direct” method. The second one is an “indirect” procedure of discrete collocation-type since we act on the so-called regularized Fredholm equation. In both cases, the convergence and the stability of the method is proved in suitable weighted spaces of functions, and the well conditioning of the linear system is showed. In order to illustrate the efficiency of the proposed procedures, some numerical tests are given.

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 Nystrom types, which are based on the zeros of suitable orthogonal polynomials. One of the main interesting aspects of our procedures regards the “quality” of the involved functions, since we can successfully manage functions which are singular on the finite boundaries and can have an exponential growth on the infinite boundaries of the domains. Moreover the errors of the methods are comparable with the error of best polynomial approximation in the weighted spaces of functions that we go to treat. The convergence and the stability of the methods and the well conditioning of the final linear systems are proved and some numerical tests, which confirm the theoretical estimates, are given.

Let w ( x ) = e − x β x α , w ¯ ( x ) = x w ( x ) and let { p m ( w ) } m , { p m ( w ¯ ) } m be the corresponding sequences of orthonormal polynomials. Since the zeros of p m + 1 ( w ) interlace those of p m ( w ¯ ) , it makes sense to construct an interpolation process essentially based on the zeros of Q 2 m + 1 : = p m + 1 ( w ) p m ( w ¯ ) , which is called “Extended Lagrange Interpolation”. In this paper the convergence of this interpolation process is studied in suitable weighted L 1 spaces, in a general framework which completes the results given by the same authors in weighted L u p ( ( 0 , + ∞ ) ) , 1 ≤ p ≤ ∞ (see [31] , [28] ). As an application of the theoretical results, an extended product integration rule, based on the aforesaid Lagrange process, is proposed in order to compute integrals of the type ∫ 0 + ∞ f ( x ) k ( x , y ) u ( x ) d x , u ( x ) = e − x β x γ ( 1 + x ) λ , γ > − 1 , λ ∈ R + , where the kernel k ( x , y ) can be of different kinds. The rule, which is stable and fast convergent, is used in order to construct a computational scheme involving the single product integration rule studied in [23] . It is shown that the “compound quadrature sequence” represents an efficient proposal for saving 1/3 of the evaluations of the function f, under unchanged speed of convergence.

In this paper we investigate some Nystr?m 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?m methods based on the so called Generalized Bernstein quadrature rule, on a Romberg scheme and on the so-called IMT rule. We prove that the proposed methods are convergent, stable and well conditioned. Also, we give several numerical tests for comparing these three methods.

The knowledge of the fluid discharge in free surface flows requires a great number of velocity measurements along the whole cross-section, taking up a large amount of time, using expensive equipment, and employing specialized labor. To overcome these obstacles, various models have been developed thus far that show how to estimate the mean velocity through the maximum velocity. In three-dimensional open channels, the maximum velocity can be located below the free surface because of the presence of secondary flows mainly originating by the sidewalls, an occurrence known as dip-phenomenon. In this condition, predicting the maximum velocity position is quite difficult and has always represented a challenge to most hydraulic engineers and researchers. In the present study, a mathematical model derived from the information entropy theory is proposed to evaluate the velocity-dip-position over the entire cross-section of both wide and narrow open channels, thus overcoming the limitations of the existing methods. Large literature measurement sets, collected in uniform and non-uniform flows, were used to test the validity of the model, showing good agreement with the experimental data and providing an accurate estimation of the dip-position.

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

Let w(x)=e−xβxα, w¯(x)=xw(x) and denote by {pm(w)}m,{pn(w¯)}n the corresponding sequences of orthonormal polynomials. The zeros of the polynomial Q2m+1=pm+1(w)pm(w¯) are simple and are sufficiently far among them. Therefore it is possible to construct an interpolation process essentially based on the zeros of Q2m+1, which is called ”Extended Lagrange Interpolation”. Here we study the convergence of this interpolation process in suitable weighted L1 spaces. This study completes the results given by the authors in previous papers in weighted Lup((0,+∞)), for 1≤p≤∞. Moreover an application of the proposed interpolation process in order to construct an e cient product quadrature scheme for weakly singular integrals is given.

The paper deals with the Lagrange interpolation of functions having a bounded variation derivative. For special systems of nodes, it is shown that this polynomial sequence converges with the best approximation order. The Lp weighted case is also discussed.

Fredholm integral equations with the right-hand side having singularities at the endpoints are considered. The singularities are moved into the kernel that is subsequently regularized by a suitable one-to-one map. The Nystrom method is applied to the regularized equation. The convergence, stability and well conditioning of the method is proved in spaces of weighted continuous functions. The special case of the weakly singular and symmetric kernel is also investigated. Several numerical tests are included.

Fredholm integral equations on the interval [−1,1] with right-hand sides having isolated singularities are considered. The original equation is reduced to an equivalent system of Fredholm integral equations with smooth input functions. The Nystrom method is applied to the system after a polynomial regularization. The convergence, stability and well conditioning of the method is proved in spaces of weighted continuous functions. The special case of the weakly singular and symmetric kernel is also investigated. Several numerical tests are included.

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

The mapping properties of the Cauchy singular integral operator with constant coefficients are studied in couples of spaces equipped with weighted uniform norms. Recently weighted Besov type spaces got more and more interest in approximation theory and, in particular, in the numerical analysis of polynomial approximation methods for Cauchy singular integral equations on an interval. In a scale of pairs of weighted Besov spaces the authors state the boundedness and the invertibility of the Cauchy singular integral operator. Such result was not expected for a long time and it will affect further investigations essentially. The technique of the paper is based on properties of the de la Vallee Poussin operator constructed with respect to some Jacobi polynomials.

The Hilbert transformation is studied on a new weighted Besov type space, characterized by means of the best approximation. This space appears as a “limit” case of the well-known Besov spaces defined by Ditzian and Totik, and it can be described with the help ofK-functionals and φ-moduli of smoothness.

Some convergent and stable numerical procedures for Cauchy singular integral equations are given. The proposed approach consists of solving the regularized equation and is based on the weighted polynomial interpolation. The convergence estimates are sharp and the obtained linear systems are well conditioned.

Convergence and boundedness of the extended Lagrange interpolating operator with additional nodes are studied in the space L_p^{u,t } of Sobolev type.

The Hilbert transform of a function g, H(g) is an important tool in many mathematical fields. Expecially its numerical evaluation is often useful in some procedures for searcing solutions of the singular integral equations. In this context an approximation of (HV^alpha,beta,f;t), |t|1, where f is a continuous function in [-1,1] and v^alpha,beta, alpha,beta>-1 is a Jacobi weight, is required. In the last decade more then one paper appeared on this subject and among others we recall [1,2,3,4,5,14,15,20]. The procedure used in these papers can be described as follows. Approximate the function f with a Lagrange interpolating polynomial and obtain a quadrature formula with coefficients depending on the singularity t. The error of the quadrature formula was estimated assuming t in a "more or less" wide neighborhood of zero since (HV^alpha,beta,f;t) is unbounded at the endpoints of (-1,1). In this paper we propose a more accurate procedure: since (HV^alpha,beta,f;t) is unbounded at the endpoints, itis more natural to consider its approximation in some functional spaces equipped with weighted uniform norms. In Section 2 (see theorems 2.1 and 2.2) we state the behaviour of (HV^alpha,beta,f;t) in [-1,1] and we give new conditions for its exstence. In sections 3 and 4 we cosntruct approximations of (HV^alpha,beta,f;t) which converge to the exact value in weighted uniform norm. The L^p convergence of the formula is also briefly treated. Finally since the considered procedures are numerically stable, these can be used for the numerical evaluation of (HV^alpha,beta,f;t) . In this sense the present paper is a short survey on the subject.

The authors study the convergence and the stability of a collocation and a discrete collocation method for Cauchy singular integral equations with weakly singular perturbation kernels in some weighted uniform norms. Uniform error estimates are also given. © 1997 Rocky Mountain Mathematics Consortium.

The behavior of the Lagrange polynomial L m (w,f) , based on the zeros of the orthogonal polynomials, is studied in some weighted Besov spaces B p r,q (u) . It is proved that L m (w) is a uniformly bounded map under suitable conditions on the weight functions and the parameters p , r , and q .