scholarly journals Multivariate Markov polynomial inequalities and Chebyshev nodes

2008 ◽  
Vol 338 (1) ◽  
pp. 350-357 ◽  
Author(s):  
Lawrence A. Harris
Keyword(s):  
Chebyshev Nodes ◽  
Polynomial Inequalities

scholarly journals On Hermite-Fejér type interpolation on the Chebyshev nodes

1993 ◽  
Vol 47 (1) ◽  
pp. 13-24 ◽  
Author(s):  
Graeme J. Byrne ◽  
T.M. Mills ◽  
Simon J. Smith
Keyword(s):  
Approximation Error ◽  
Interpolation Polynomial ◽  
Precise Estimate ◽  
Type Interpolation ◽  
Interpolation Methods ◽  
Chebyshev Nodes ◽  
Interpolatory Process

Given f ∈ C [−1, 1], let Hn, 3(f, x) denote the (0,1,2) Hermite-Fejér interpolation polynomial of f based on the Chebyshev nodes. In this paper we develop a precise estimate for the magnitude of the approximation error |Hn, 3(f, x) − f(x)|. Further, we demonstrate a method of combining the divergent Lagrange and (0,1,2) interpolation methods on the Chebyshev nodes to obtain a convergent rational interpolatory process.

scholarly journals Polynomial inequalities on the Hamming cube

2020 ◽  
Vol 178 (1-2) ◽  
pp. 235-287
Author(s):  
Alexandros Eskenazis ◽  
Paata Ivanisvili
Keyword(s):  
Polynomial Inequalities

scholarly journals Lebesgue constants for polyhedral sets and polynomial interpolation on Lissajous–Chebyshev nodes

2017 ◽  
Vol 43 ◽  
pp. 1-27 ◽  
Author(s):  
Peter Dencker ◽  
Wolfgang Erb ◽  
Yurii Kolomoitsev ◽  
Tetiana Lomako
Keyword(s):  
Polynomial Interpolation ◽  
Lebesgue Constants ◽  
Polyhedral Sets ◽  
Chebyshev Nodes

scholarly journals Optimal sampling rates for approximating analytic functions from pointwise samples

2018 ◽  
Vol 39 (3) ◽  
pp. 1360-1390 ◽  
Author(s):  
Ben Adcock ◽  
Rodrigo B Platte ◽  
Alexei Shadrin
Keyword(s):  
Least Squares ◽  
Analytic Functions ◽  
Exponential Convergence ◽  
Sampling Rate ◽  
Arbitrary Point ◽  
Impossibility Theorem ◽  
Weight Functions ◽  
Remez Algorithm ◽  
Chebyshev Nodes ◽  
Methods Numerical

AbstractWe consider the problem of approximating an analytic function on a compact interval from its values at $M+1$ distinct points. When the points are equispaced, a recent result (the so-called impossibility theorem) has shown that the best possible convergence rate of a stable method is root-exponential in M, and that any method with faster exponential convergence must also be exponentially ill conditioned at a certain rate. This result hinges on a classical theorem of Coppersmith & Rivlin concerning the maximal behavior of polynomials bounded on an equispaced grid. In this paper, we first generalize this theorem to arbitrary point distributions. We then present an extension of the impossibility theorem valid for general nonequispaced points and apply it to the case of points that are equidistributed with respect to (modified) Jacobi weight functions. This leads to a necessary sampling rate for stable approximation from such points. We prove that this rate is also sufficient, and therefore exactly quantify (up to constants) the precise sampling rate for approximating analytic functions from such node distributions with stable methods. Numerical results—based on computing the maximal polynomial via a variant of the classical Remez algorithm—confirm our main theorems. Finally, we discuss the implications of our results for polynomial least-squares approximations. In particular, we theoretically confirm the well-known heuristic that stable least-squares approximation using polynomials of degree N < M is possible only once M is sufficiently large for there to be a subset of N of the nodes that mimic the behavior of the $N$th set of Chebyshev nodes.

scholarly journals Explicit Bounds for Hermite Polynomials in the Oscillatory Region

2000 ◽  
Vol 3 ◽  
pp. 307-314 ◽  
Author(s):  
William H. Foster ◽  
Ilia Krasikov
Keyword(s):  
Quadratic Forms ◽  
Hermite Polynomials ◽  
Polynomial Inequalities ◽  
Explicit Bounds

AbstractWe apply a method of positive quadratic forms based on polynomial inequalities to establish sharp explicit bounds on the envelope of Hermite polynomials in the oscillatory region |x| < (2k – 3/2)1/2.

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