Asymptotic expansion and extrapolation for Bernstein polynomials with applications

Costabile, F.; Gualtieri, Maria Italia; Serra, Stefano

doi:10.1007/bf01733787

Cited by 27 publications

(22 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…-Theorem 4.2 provides a constructive tool to build up linear positive operators for the approximation of continuous functions with any number of variables. The use of extrapolation methods to accelerate the related rate of convergence can be also considered [8,40,21,22]. Moreover other approximation problems of rational types can be treated [55]: the main tools are the LPOs devised in Sect.…”

Section: Some Consequencesmentioning

confidence: 99%

A Korovkin-type theory for finite Toeplitz operators via matrix algebras

Serra¹

1999

Numerische Mathematik

View full text Add to dashboard Cite

Preconditioned conjugate gradients (PCG) are widely and successfully used methods for solving a Toeplitz linear system A n x = b [59,9,20,5,34,62,6,10,28,45,44,46,49]. Frobenius-optimal preconditioners are chosen in some proper matrix algebras and are defined by minimizing the Frobenius distance from A n . The convergence features of these PCG have been naturally studied by means of the Weierstrass-Jackson Theorem [17,36,49], owing to the profound relationship between the spectral features of the matrices A n , generated by the Fourier coefficients of a continuous function f , and the analytical properties of the symbol f itself. In this paper, we capsize this point of view by showing that the optimal preconditioners can be used to define both new and just known linear positive operators uniformly approximating the function f . On the other hand, by modifying the Korovkin Theorem to study the Frobenius-optimal preconditioning problem, we provide a new and unifying tool for analyzing all Frobenius-optimal preconditioners in any generic matrix algebra related to trigonometric transforms. Finally, the multilevel case is sketched and discussed by showing that a Korovkin-type Theory also holds in a multivariate sense. Subject Classification (1991): 65F10, 47B25, 41A36, 15A18 Mathematics

show abstract

Section: Some Consequencesmentioning

confidence: 99%

A Korovkin-type theory for finite Toeplitz operators via matrix algebras

Serra¹

1999

Numerische Mathematik

View full text Add to dashboard Cite

show abstract

“…The correspondence between functions and sequences of nested Toeplitz matrices defined by Fourier coefficients suggests us that a natural, but not compulsory choice of the spaces V n is given by the trigonometric polynomials of degree at most n. When we consider symmetric Toeplitz matrices, the associated generating function is also even and therefore, in this case, it is natural to define V n as the space of even trigonometric polynomials. Let us define {S n : C 2π → V n } n as a sequence of linear approximation operators [25,18,11,12] and let A n ∈ A T be a fixed n × n trigonometric matrix algebra. With these notations we indicate by A n (S n (f )) the matrix belonging to the algebra A n such that…”

Section: The Construction Of the Preconditionersmentioning

confidence: 99%

A unifying approach to abstract matrix algebra preconditioning

Benedetto

Serra‐Capizzano

1999

Numerische Mathematik

View full text Add to dashboard Cite

In earlier papers Tyrtyshnikov [42] and the first author [14] considered the analysis of clustering properties of the spectra of specific Toeplitz preconditioned matrices obtained by means of the best known matrix algebras.Here we generalize this technique to a generic Banach algebra of matrices by devising general preconditioners related to "convergent" approximation processes [36].Finally, as case study, we focus our attention on the Tau preconditioning by showing how and why the best matrix algebra preconditioners for symmetric Toeplitz systems can be constructed in this class. Classification (1991): 65F10, 65F35 Mathematics Subject

show abstract

“…Moreover, if a sequence {(J n} possesses an expansion of the type (1) for all m E llV , then we say that the expansion is of arbitrary order, and write (2) for short.…”

Section: Jl=1mentioning

confidence: 99%

Asymptotic expansions for multivariate polynomial approximation

Walz

2000

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

In this paper the approximation of multivariate functions by (multivariate) Bernstein polynomials is considered. Building on recent work of Lai [3], we can prove that the sequence of these Bernstein polynomials possesses an asymptotic expansion with respect to the index n. This generalizes a corresponding result due to Costabile, Gualtieri and Serra [2] on univariate Bernstein polynomials, providing at the same time a new proof for it. After having shown the existence of an asymptotic expansion we can apply an extrapolation algorithm which accelerates the convergence of the Bernstein polynomials considerably; this leads to a new and very efficient method for polynomial approximation of multivariate functions. Numerical examples illustrate our approach.

show abstract

Asymptotic expansion and extrapolation for Bernstein polynomials with applications

Cited by 27 publications

References 5 publications

A Korovkin-type theory for finite Toeplitz operators via matrix algebras

A Korovkin-type theory for finite Toeplitz operators via matrix algebras

A unifying approach to abstract matrix algebra preconditioning

Asymptotic expansions for multivariate polynomial approximation

Contact Info

Product

Resources

About