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Near-best approximation by averaging polynomial interpolants. (English) Zbl 0621.41005
Let A(C) be the space of functions which are analytic in an open unit disk and continuous on the unit circle. Let also $$\prod_{n-1}\subset A(C)$$ be the subspace of polynomial functions of degree less than n. Using the sup-norm on A(C) one can consider various polynomial interpolation techniques which give rise to (bounded, linear) projection operators $$p: A(C)\to \prod_{n-1}.$$ Clearly $$\| f-pf\| \leq (1+\| p\|)(\| f-q\|)$$ for any $$q\in \prod_{n-1}$$. Hence, any such projection provides a so-called near-best approximation within a relative distance $$\| p\|$$. Let $$\omega_{q,j}=\exp (2\pi i(jm- q)/mn),$$ $$q=0,...,m-1$$; $$j=1,...,n$$, be the set of mnth roots of unity. Then the Fourier projection $$F_{n-1,q}: A(C)\to \prod_{n-1}$$ is given by the polynomial interpolation of f in the points $$\{\omega_{q,j}\}$$, $$j=1,...,n$$ (q fixed). The norm $$\| F_{n-1,q}\|$$ was calculated by T. H. Gronwall [Bull. Am. Math. Soc. 27, 275-279 (1921)].
The authors introduce the mean projection $$M_{n-1,m}$$ by the formula $M_{n-1,m}(f)=(1/m)\sum^{m-1}_{q=0}F_{n-1,q}(f)$ and calculate its norm. These calculations are used in order to show that $$\lim_{m\to \infty}M_{n-1,m}$$ is equal to the projection of functions to the first n terms of their Taylor series. This result is interesting due to the fact that Taylor projection has minimal norm among all the projections from A(C) to $$\prod_{n-1}$$ [cf. K. O. Geddes and J. C. Mason [SIAM J. Numer. Anal. 12, 111-120 (1975; Zbl 0271.65005)]. Some other results on polynomial interpolation are also presented in the paper under review.
Reviewer: A.Kushkuley

MSC:
 41A10 Approximation by polynomials 41A05 Interpolation in approximation theory 42A10 Trigonometric approximation 42A15 Trigonometric interpolation 41A50 Best approximation, Chebyshev systems
Zbl 0271.65005
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