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Interpolating them-th power ofx at the zeros of then-th Chebyshev-polynomial yields an almost best Chebyshev-approximation (1986)

Fiedler, H.

Springer

Aequationes mathematicae 31 (1986), S. 294-299

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ISSN: 1420-8903

Keywords: 41A10, 41A50

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: Abstract LetX={x 1,x 2,..., n }εI=[−1, 1] and $$\varphi (x) = \prod\limits_{j = 1}^n {(x - x_j )} $$ . Forf∈C 1(I) definef* byf−p f =φf*, wherep f denotes the interpolation-polynomial off with respect toX. We state some properties of the operatorf →f*. In particular, we treat the case whereX consists of the zeros of the Chebyshev polynomialT n (x) and obtain ∥x m −p x m∥≤8eE n−1(x m ), whereE n−1(f) denotes the sup-norm distance fromf to the polynomials of degree less thann. Finally we state a lower estimate forE n (f) that omits theassumptionf (n+1)〉0 in a similar estimate of Meinardus.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF02188195

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