scholarly journals Hermite-Fejer interpolation at the ‘practical’ Chebyshev nodes

1973 ◽  
Vol 9 (3) ◽  
pp. 379-390
Author(s):  
R.D. Riess
Keyword(s):  
Continuous Functions ◽  
Negative Results ◽  
Chebyshev Nodes ◽  
Chebyshev Points ◽  
Uniformly Convergent ◽  
Lagrangian Interpolation

Berman has raised the question in his work of whether Hermite-Fejér interpolation based on the so-called “practical” Chebyshev points, , 0(1)n, is uniformly convergent for all continuous functions on the interval [−1, 1]. In spite of similar negative results by Berman and Szegö, this paper shows this result is true, which is in accord with the great similarities of Lagrangian interpolation based on these points versus the points , 1(1)n.

scholarly journals On the unconditional convergence of wavelet expansions for continuous functions

2016 ◽  
Vol 14 (01) ◽  
pp. 1650007 ◽  
Author(s):  
Naohiro Fukuda ◽  
Tamotu Kinoshita ◽  
Toshio Suzuki
Keyword(s):  
Continuous Functions ◽  
Unconditional Convergence ◽  
Wavelet Expansions ◽  
Counter Example ◽  
Uniformly Convergent

In this paper, we study the unconditional convergence of wavelet expansions with Lipschitz wavelets. Especially with the Strömberg wavelet, we shall construct a counter example which shows that uniformly convergent wavelet expansions even for continuous functions do not always converge unconditionally in [Formula: see text].

scholarly journals A Universal Approximation Theorem for Mixture-of-Experts Models

2016 ◽  
Vol 28 (12) ◽  
pp. 2585-2593 ◽  
Author(s):  
Hien D. Nguyen ◽  
Luke R. Lloyd-Jones ◽  
Geoffrey J. McLachlan
Keyword(s):  
Network Architecture ◽  
Approximation Theorem ◽  
Target Function ◽  
Continuous Functions ◽  
Universal Approximation ◽  
Mixture Of Experts ◽  
Neural Network Architecture ◽  
Alternative Result ◽  
Unit Hypercube ◽  
Uniformly Convergent

The mixture-of-experts (MoE) model is a popular neural network architecture for nonlinear regression and classification. The class of MoE mean functions is known to be uniformly convergent to any unknown target function, assuming that the target function is from a Sobolev space that is sufficiently differentiable and that the domain of estimation is a compact unit hypercube. We provide an alternative result, which shows that the class of MoE mean functions is dense in the class of all continuous functions over arbitrary compact domains of estimation. Our result can be viewed as a universal approximation theorem for MoE models. The theorem we present allows MoE users to be confident in applying such models for estimation when data arise from nonlinear and nondifferentiable generative processes.

scholarly journals A new approach for constructing mock-Chebyshev grids

2021 ◽  
Author(s):  
Ali IBRAHIMOGLU
Keyword(s):  
Numerical Experiments ◽  
Polynomial Interpolation ◽  
Numerical Results ◽  
Uniform Grid ◽  
New Approach ◽  
Chebyshev Nodes ◽  
Chebyshev Points ◽  
Runge Phenomenon ◽  
Equidistant Nodes

Polynomial interpolation with equidistant nodes is notoriously unreliable due to the Runge phenomenon, and is also numerically ill-conditioned. By taking advantage of the optimality of the interpolation processes on Chebyshev nodes, one of the best strategies to defeat the Runge phenomenon is to use the mock-Chebyshev points, which are selected from a satisfactory uniform grid, for polynomial interpolation. Yet, little literature exists on the computation of these points. In this study, we investigate the properties of the mock-Chebyshev nodes and propose a subsetting method for constructing mock-Chebyshev grids. Moreover, we provide a precise formula for the cardinality of a satisfactory uniform grid. Some numerical experiments using the points obtained by the method are given to show the effectiveness of the proposed method and numerical results are also provided.

Weighted approximation via Θ-summations of Fourier-Jacobi series

2010 ◽  
Vol 47 (2) ◽  
pp. 139-154
Author(s):  
Ágnes Chripkó
Keyword(s):  
Banach Space ◽  
Order Of Convergence ◽  
Weighted Approximation ◽  
Continuous Functions ◽  
Gamma Delta ◽  
Jacobi Series ◽  
Jacobi Weights ◽  
Special Cases ◽  
Uniformly Convergent ◽  
De La Vallée Poussin

This paper is devoted to the study of Θ-summability of Fourier-Jacobi series. We shall construct such processes (using summations) that are uniformly convergent in a Banach space (\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$C_{w_{\gamma ,\delta } } ,\parallel \cdot \parallel _{w_{\gamma ,\delta } }$$ \end{document}) of continuous functions. Some special cases are also considered, such as the Fejér, de la Vallée Poussin, Cesàro, Riesz and Rogosinski summations. Our aim is to give such conditions with respect to Jacobi weights wγ,δ , wα,β and to summation matrix Θ for which the uniform convergence holds for all f ∈ \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$C_{w_{\gamma ,\delta } }$$ \end{document}. Order of convergence will also be investigated. The results and the methods are analogues to the discrete case (see [16] and [17]).

scholarly journals On certain non-archimedean functions analogous to complex analytic functions

1976 ◽  
Vol 14 (1) ◽  
pp. 23-36 ◽  
Author(s):  
Kurt Mahler
Keyword(s):  
Power Series ◽  
Analytic Functions ◽  
Convergent Series ◽  
Continuous Functions ◽  
Algebraic Extension ◽  
Finite Degree ◽  
Uniformly Convergent ◽  
Image Position ◽  
Derivatives Of ◽  
Special Case

Let Qp and K be the rational p–adic field and an algebraic extension of Qp of finite degree, respectively, and let I and Ik be the subsets of Qp and of K consisting of the p–adic integers of these fields.It is known that the continuous functions f: I → Qp can be written aswhere this series converges uniformly on I, and that such continuous functions need not be differentiable at any point. We here study continuous functions F: IK → K which for all X on IK are the sum of a uniformly convergent seriesIt is proved that such functions F(X) have at every point of IK derivatives of all orders. In the special case when K is totally ramified, they cannot in general be developed into power series that converge everywhere on IK, but this is possible when K is not totally ramified.

Tables for Lagrangian Interpolation using Chebyshev Points.

1982 ◽  
Vol 39 (160) ◽  
pp. 743
Author(s):  
Herbert E. Salzer ◽  
Norman Levine ◽  
Saul Serben
Keyword(s):  
Chebyshev Points ◽  
Lagrangian Interpolation

scholarly journals Uniformly summing sets of operators on spaces of continuous functions

2004 ◽  
Vol 2004 (63) ◽  
pp. 3397-3407
Author(s):  
J. M. Delgado ◽  
Cándido Piñeiro
Keyword(s):  
Banach Spaces ◽  
Continuous Functions ◽  
General Properties ◽  
Spaces Of Continuous Functions ◽  
Uniformly Convergent

LetXandYbe Banach spaces. A setℳof 1-summing operators fromXintoYis said to beuniformly summingif the following holds: given a weakly 1-summing sequence(xn)inX, the series∑n‖Txn‖is uniformly convergent inT∈ℳ. We study some general properties and obtain a characterization of these sets whenℳis a set of operators defined on spaces of continuous functions.

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