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A note on Hermite-Fejér interpolation for the unit circle. (English) Zbl 0982.41003

For every pair (p,q) of integers, where pq we denote by Λ p,q the linear space of all Laurent polynomials (L-polynomials)

L(z)= j=p q c j z j ,c j ·

Suppose that p(n), q(n) are two nondecreasing sequences of nonnegative integers such that p(n)+q(n)=2n-1, n=1,2, with lim n (n)=lim n q(n)=, and |p(n)-n| bounded. In this paper the following result is proved:

Theorem. Let f be a continuous function on 𝕋={z:|z|=1} and let L n be the unique L-polynomial in Λ -p(n),q(n) satisfying

L n (z k )=f(z k ),L n ' (z k )=0,k=1,2,,n

where z k are the roots of z n +λ n =0, {λ n } being an arbitrary sequence on 𝕋. Then the sequence L n converges to f uniformly on 𝕋.

This result is an extension to the unit circle T of the classical Hermite-Fejér theorem about an approximation on the interval [-1,1].

MSC:
41A10Approximation by polynomials
41A17Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
30C10Polynomials (one complex variable)