Near-best approximation by averaging polynomial interpolants.

*(English)*Zbl 0621.41005Let 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.

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 |