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Distribution of interpolation points of best \(L_ 2\)-approximants (\(n\)th partial sums of Fourier series). (English) Zbl 0780.41003

Summary: For a continuous \(2\pi\)-periodic real-valued function \(f\), we investigate the asymptotic behavior of the zeros of the error \(f(\theta)-s_ n(\theta)\), where \(s_ n(\theta)\) is the \(n\)th Fourier section. We prove that there is a subsequence \(\{n_ k\}\) for which such zeros (interpolation points) are uniformly distributed on \([-\pi,\pi]\). This extends previous results of Saff and Shekhtman. Moreover, results dealing with the maximal distance between consecutive zeros of \(f-s_{n_ k}\) are obtained. The technique of proof involves coefficient estimates for lacunary trigonometric polynomials in terms of its \(L_ q\)-norm on a subinterval.

MSC:

41A05 Interpolation in approximation theory
41A50 Best approximation, Chebyshev systems
42A10 Trigonometric approximation
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