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The Lebesgue Constant for Higher Order Hermite-Fejér Interpolation on the Chebyshev Nodes
Journal of Approximation Theory, 1995 Let \(\lambda_{2m, n} (x)\) denote the Lebesgue function associated with \((0, 1,\dots, 2m)\) Hermite-Fejér polynomial interpolation on the Chebyshev nodes \(x_{k,n}:= \text{cos} [(2k-1) \pi/2n]\), \(k=1, \dots, n\). Here \(m\geq 0\) and \(n\) runs over positive integers.
Graeme Byrne +2 more
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A note on polynomial interpolation at the Chebyshev extrema nodes
Journal of Approximation Theory, 1984 Some new properties of the Lebesgue function associated with interpolation at the Chebyshev extrema nodes are established. By estimating the degree of asymmetry of the Lebesgue function in the interval of interest, the estimate of Ehlich and Zeller for the norm of the corresponding interpolation operator is improved.
L. Brutman
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Approximation by interpolation: the Chebyshev nodes [PDF]
Journal of Classical Analysis, 2020 M. Foupouagnigni +3 more
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The minimal number of nodes in Chebyshev type quadrature formulas
Indagationes Mathematicae, 1993 The authors abstract is accurate: We study Chebyshev type quadrature formulas of degree \(n\) with respect to a weight function on \(\langle - 1,+1 \rangle\), i.e. formulas \[ {1 \over {\int_{-1}^{+1} w(t)dt}}\cdot \int_{-1}^{+1} f(t) w(t) dt={1\over N} \sum_{i=1}^ N f(x_ i)+ R(f) \] with nodes \(x_ i\in \langle -1,+1\rangle\) such that \(R(f)=0\) for ...
Arno B. J. Kuijlaars
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Bivariate Lagrange interpolation at the Chebyshev nodes
Proceedings of the American Mathematical Society, 2010 We discuss Lagrange interpolation on two sets of nodes in two dimensions where the coordinates of the nodes are Chebyshev points having either the same or opposite parity. We use a formula of Xu for Lagrange polynomials to obtain a general interpolation theorem for bivariate polynomials at either set of Chebyshev nodes.
Lawrence A. Harris
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A polynomial interpolation process at quasi-Chebyshev nodes with the FFT [PDF]
Mathematics of Computation, 2011 Interpolation polynomial p n p_n at the Chebyshev nodes cos π j / n \cos \pi j/n ( 0 ≤ j ≤ n 0\le j\le n ) for smooth functions is known to converge fast as n →
Hiroshi Sugiura, Takemitsu Hasegawa
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Dimension reduction of thermoelectric properties using barycentric polynomial interpolation at Chebyshev nodes. [PDF]
Sci Rep, 2020 Chung J, Ryu B, Park S.
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The Lebesgue function for Lagrange interpolation on the augmented Chebyshev nodes
Publicationes Mathematicae Debrecen, 2005 Summary: Given \(f\in C[-1,1]\) and \(n\) points (nodes) in \([-1,1]\), the well-known Lagrange interpolation polynomial is the polynomial of minimum degree which agrees with \(f\) at each of the nodes. Properties of the Lebesgue function and Lebesgue constant associated with Lagrange interpolation on the Chebyshev nodes (the zeros of the \(n\)th ...
Simon J. Smith
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Alternating polynomials associated with the Chebyshev extrema nodes
Journal of Approximation Theory, 1988 For a given set \(X=\{x_ k\}_ 0^{n+1}\) of nodes in [-1,1], the author introduces an operator \(A_ n(X;x)\) mapping functions in C[-1,1] into polynomials of degree n. They were called `generalized alternating polynomials'. Here X is specialized to \(U=\{\cos k\pi /n+1\}_ 0^{n+1}\), the extrema of Chebyshev polynomials of the first kind.
L. Brutman
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A fast algorithm for computing the mock-Chebyshev nodes
Journal of Computational and Applied Mathematics, 2019 Abstract Runge Phenomenon which is a very well-known example and published by C. Runge in 1901 is as follows: polynomial interpolation of a function f , using equidistant interpolation points on [ − 1 , 1 ] could diverge on certain parts of this interval even if f is analytic anywhere on the interval.
B. Ali Ibrahimoglu
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