296 research outputs found
Some remarks on filtered polynomial interpolation at Chebyshev nodes
The 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
On the filtered polynomial interpolation at Chebyshev nodes
The paper deals with a special filtered approximation method, which
originates interpolation polynomials at Chebyshev zeros by using de la Vall\'ee
Poussin filters. These polynomials can be an useful device for many theoretical
and applicative problems since they combine the advantages of the classical
Lagrange interpolation, with the uniform convergence in spaces of locally
continuous functions equipped with suitable, Jacobi--weighted, uniform norms.
The uniform boundedness of the related Lebesgue constants, which equals to the
uniform convergence and is missing from Lagrange interpolation, has been
already proved in literature under different, but only sufficient, assumptions.
Here, we state the necessary and sufficient conditions to get it. These
conditions are easy to check since they are simple inequalities on the
exponents of the Jacobi weight defining the norm. Moreover, they are necessary
and sufficient to get filtered interpolating polynomials with a near best
approximation error, which tends to zero as the number of nodes tends to
infinity. In addition, the convergence rate is comparable with the error of
best polynomial approximation of degree , hence the approximation order
improves with the smoothness of the sought function. Several numerical
experiments are given in order to test the theoretical results, to make a
comparison with the Lagrange interpolation at the same nodes and to show how
the Gibbs phenomenon can be strongly reduced.Comment: 20 pages, 19 figures given in 8 eps file
A generalization of Floater--Hormann interpolants
In this paper the interpolating rational functions introduced by Floater and
Hormann are generalized leading to a whole new family of rational functions
depending on , an additional positive integer parameter. For , the original Floater--Hormann interpolants are obtained. When we
prove that the new rational functions share a lot of the nice properties of the
original Floater--Hormann functions. Indeed, for any configuration of nodes,
they have no real poles, interpolate the given data, preserve the polynomials
up to a certain fixed degree, and have a barycentric-type representation.
Moreover, we estimate the associated Lebesgue constants in terms of the minimum
() and maximum () distance between two consecutive nodes. It turns out
that, in contrast to the original Floater-Hormann interpolants, for all we get uniformly bounded Lebesgue constants in the case of equidistant and
quasi-equidistant nodes configurations (i.e., when ). In such cases,
we also estimate the uniform and the pointwise approximation errors for
functions having different degree of smoothness.
Numerical experiments illustrate the theoretical results and show a better
error profile for less smooth functions compared to the original
Floater-Hormann interpolants.Comment: 29 page
Image Scaling by de la Vallée-Poussin Filtered Interpolation
We present a new image scaling method both for downscaling and upscaling, running with any scale factor or desired size. The resized image is achieved by sampling a bivariate polynomial which globally interpolates the data at the new scale. The methodâs particularities lay in both the sampling model and the interpolation polynomial we use. Rather than classical uniform grids, we consider an unusual sampling system based on Chebyshev zeros of the first kind. Such optimal distribution of nodes permits to consider near-best interpolation polynomials defined by a filter of de la VallĂ©e-Poussin type. The action ray of this filter provides an additional parameter that can be suitably regulated to improve the approximation. The method has been tested on a significant number of different image datasets. The results are evaluated in qualitative and quantitative terms and compared with other available competitive methods. The perceived quality of the resulting scaled images is such that important details are preserved, and the appearance of artifacts is low. Competitive quality measurement values, good visual quality, limited computational effort, and moderate memory demand make the method suitable for real-world applications
Filtered integration rules for finite weighted Hilbert transforms
A product quadrature rule, based on the filtered de la VallĂ©e Poussin polynomial approximation, is proposed for evaluating the finite weighted Hilbert transform in [â1,1]. Convergence results are stated in weighted uniform norm for functions belonging to suitable Besov-type subspaces. Several numerical tests are provided, also comparing the rule with other formulas known in literature
Some numerical applications of generalized Bernstein operators
Inthispaper,somerecentapplicationsoftheso-calledGeneralizedBernsteinpolynomialsarecollected. 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 can be suitable chosen in order to improve the rate of convergence to the function f , as the smoothness of f increases, overcoming the well-known low degree of approximation achieved by the classical Bernstein polynomials or by the 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 em- ployed in NystroÌm 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
Real-practice thromboprophylaxis in atrial fibrillation
This retrospective observational study was based on databases of the Local Health Authority of Treviso, Italy. It evaluated the prevalence and the effectiveness of oral anticoagulation treatment (OAT) for the management of nonvalvular atrial fibrillation (NVAF) in everyday clinical practice. Out of 6,138 NVAF patients, only 3,024 received Vitamin K antagonist (VKA). Potential barriers decreasing the probability of being treated with VKA were female sex, older age, antiplatelet treatment and history of bleeding. In addition, VKA-treatment was not in line with current ESC and AIAC guidelines, since the patients at high or low risk of stroke were under-or over-treated, resp. Among VKAtreated patients, 73 % of subjects were not at target with anticoagulation. OAT resulted to be effective in reducing stroke risk. However, stroke events were significantly influenced also by previous stroke or transient ischemic attack (hazard ratio, HR = 2.99, p < 0.001) and by previous bleeding events (HR = 1.60, p < 0.001)
Role of transoesophageal echocardiography in evaluating the effect of catheter ablation of atrial fibrillation on anatomy and function of the pulmonary veins
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