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On the Positivity of the Fundamental Polynomials for Generalized Hermite–Fejér Interpolation on the Chebyshev Nodes

✍ Scribed by Simon J. Smith

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
90 KB
Volume
96
Category
Article
ISSN
0021-9045

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✦ Synopsis

It is shown that the fundamental polynomials for (0, 1, ..., 2m+1) Hermite Feje r interpolation on the zeros of the Chebyshev polynomials of the first kind are nonnegative for &1 x 1, thereby generalising a well-known property of the original Hermite Feje r interpolation method. As an application of the result, Korovkin's theorem on monotone operators is used to present a new proof that the (0, 1, ..., 2m+1) Hermite Feje r interpolation polynomials of f # C[&1, 1], based on n Chebyshev nodes, converge uniformly to f as n Ä .

1999 Academic Press H (r) m, n (X, f, x k, n )=$ 0, r f (x k, n ), 1 k n, 0 r m.

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