1,721,005 research outputs found
Approximation of the Hilbert transform on the half–line
Full text linkThe paper concerns the weighted Hilbert transform of locally continuous functions on the semiaxis. By using a filtered de la Vallée Poussin type approximation polynomial recently introduced by the authors, it is proposed a new “truncated” product quadrature rule (VP- rule). Several error estimates are given for different smoothness degrees of the integrand ensuring the uniform convergence in Zygmund and Sobolev spaces. Moreover, new estimates are proved for the weighted Hilbert transform and for its approximation (L-rule) by means of the truncated Lagrange interpolation at the same Laguerre zeros. The theoretical results are validated by the numerical experiments that show a better performance of the VP-rule versus the L-rule
Weakly singular linear Volterra integral equations: A Nyström method in weighted spaces of continuous functions
Full text linkThis 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
Uniform weighted approximation on the square by polynomial interpolation at Chebyshev nodes
No full textThe 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 conditions are proved for the uniform boundedness of the related Lebesgue constants. Error estimates in some Sobolev-type spaces are also given. Pros and cons of such a kind of filtered interpolation are analyzed in comparison with the Lagrange polynomials interpolating at the same Chebyshev grid or at the equal number of Padua nodes. The advantages in reducing the Gibbs phenomenon are shown by means of some numerical experiments
A projection method with smoothing transformation for second kind volterra integral equations
Full text linkIn this paper we present a projection method for linear second kind Volterra integral equations with kernels having weak diagonal and/or boundary singularities of algebraic type. The proposed approach is based on a specific optimal interpolation process and a smoothing transformation. The convergence of the method is proved in suitable spaces of functions, equipped with the uniform norm. Several tests show the accuracy of the presented method
On the filtered polynomial interpolation at Chebyshev nodes
No full textThe 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 previous works this has already been proved under different sufficient conditions. Here, we complete the study by stating also the necessary conditions to get it. Several numerical experiments are also given to test the theoretical results and make comparisons to Lagrange interpolation at the same nodes
Compounded Product Integration rules on (0, +∞)
Full text linkThe paper deals with the approximation of integrals of the type I(f , y) =: \int_{0}^{+\infty} f (x)k(x, y)\rho(x) dx, \rho(x) := e^{−x} x^\gamma , where f is a sufficiently smooth function and the kernel k collects criticisms of many different types (highly oscillating, weakly singular, "nearly" singular, etc.). We propose an extended product rule based on the approximation of f by an extended Lagrange process at Laguerre zeros. We prove that the rule is stable and convergent with order of the best polynomial approximation in suitable function spaces. Furthermore, by combining the stated rule with a related product formula, we define a pattern that allows a significant saving in number of function evaluations. We give details on the construction of the coefficients of the rule for some selected kernels. Finally, some numerical tests are proposed to show the efficiency of the compounded quadrature scheme
Some remarks on filtered polynomial interpolation at Chebyshev nodes
Full text linkThe present paper concerns filtered de la Vallée 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 (uniform boundedness of the Lebesgue constants and reduction of the Gibbs phenomenon). Here we focus on some additional features that are useful in the applications of filtered VP interpolation. In particular, we analyze the simultaneous approximation provided by the derivatives of the VP interpolation polynomials. Moreover, we state the uniform boundedness of VP approximation operators in some Sobolev and Hölder-Zygmund spaces where several integro-differential models are uniquely and stably solvable
A mixed scheme of product integration rules in (−1,1)
Full text linkThe paper deals with the numerical approximation of integrals of the type I(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 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
On the numerical solution of Volterra integral equations on equispaced nodes
Full text linkIn 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. Numerical tests illustrate the performance of the proposed approach
A projection method for Volterra integral equations in weighted spaces of continuous functions
No full textThis paper is concerned with the numerical treatment of second kind Volterra integral equations whose integrands present diagonal and/or endpoint algebraic singularities. A projection method based on an optimal interpolating operator is developed in the spaces of weighted continuous functions endowed with the supremum norm. In such spaces, the uniqueness of the solution is discussed and suitable conditions are determined to assure the stability and the convergence of the method. Several numerical tests are presented to show the efficiency of the method and the agreement with the theoretical estimates
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