On the Erdös Conjecture Concerning Minimal Norm Interpolation on the Unit Circle

Abstract

Previous article Next article On the Erdös Conjecture Concerning Minimal Norm Interpolation on the Unit CircleL. Brutman and A. PinkusL. Brutman and A. Pinkushttps://doi.org/10.1137/0717031PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractThe purpose of this note is to complete the proof of the Erdös conjecture concerning optimal points of polynomial interpolation on the unit circle when the number of these points is even.[1] Carl de Boor and , Allan Pinkus, Proof of the conjectures of Bernstein and Erdo˝s concerning the optimal nodes for polynomial interpolation, J. Approx. Theory, 24 (1978), 289–303 10.1016/0021-9045(78)90014-X 80d:41003 0412.41002 CrossrefISIGoogle Scholar[2] L. Brutman, On the polynomial and rational projections in the complex plane, SIAM J. Numer. Anal., 17 (1980), 366–372 10.1137/0717030 82b:30043 0506.30025 LinkISIGoogle Scholar[3] P. Erdös, Problems and results on the convergence and divergence properties of the Lagrange interpolation polynomials and some extremal problems, Mathematica (Cluj), 10 (33) (1967), 65–73 38:1437 Google Scholar[4] Theodore A. Kilgore, A characterization of the Lagrange interpolating projection with minimal Tchebycheff norm, J. Approx. Theory, 24 (1978), 273–288 10.1016/0021-9045(78)90013-8 80d:41002 0428.41023 CrossrefISIGoogle Scholar[5] A. H. Tureckiinodoť, Theory of Interpolation in Problem Form, Izdat. Vyšeišaja Škola, Minsk, 1968, (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Rootfinding and Function EvaluationA Graduate Introduction to Numerical Methods | 21 November 2013 Cross Ref Classical (Unweighted) and Weighted InterpolationA Panorama of Hungarian Mathematics in the Twentieth Century I | 1 Jan 2006 Cross Ref Computation of the Riesz-Herglotz transform and its application to quadrature formulas over the unit circleNumerische Mathematik, Vol. 93, No. 1 | 1 Nov 2002 Cross Ref Blind identification of multichannel FIR systems based on linear predictionIEEE Transactions on Signal Processing, Vol. 48, No. 9 | 1 Sep 2000 Cross Ref Paul Erdős (1913–1996)Journal of Approximation Theory, Vol. 94, No. 1 | 1 Jul 1998 Cross Ref A Minimax Problem Admitting the Equioscillation Characterization of Bernstein and ErdősJournal of Approximation Theory, Vol. 92, No. 3 | 1 Mar 1998 Cross Ref Estimation of model qualityAutomatica, Vol. 31, No. 12 | 1 Dec 1995 Cross Ref Polynomial Interpolation on Quasi-Equidistributed Nodes on the Unit DiskHiroshi Sugiura and Tatsuo ToriiSIAM Journal on Numerical Analysis, Vol. 29, No. 4 | 14 July 2006AbstractPDF (1151 KB)Polynomial Extrapolation from $[ - 1,1]$ to the Unit DiscL. BrutmanSIAM Journal on Numerical Analysis, Vol. 28, No. 2 | 14 July 2006AbstractPDF (549 KB)The Fourier operator of even order and its application to an extremum problem in interpolationAlgorithms for Approximation II | 1 Jan 1990 Cross Ref Optimal interpolation in a general setting—A beginningJournal of Approximation Theory, Vol. 55, No. 3 | 1 Dec 1988 Cross Ref Properties of projections obtained by averaging certain polynomial interpolantsApproximation Theory, Tampa | 19 September 2006 Cross Ref On the optimal Lebesgue constants for polynomial interpolationActa Mathematica Hungarica, Vol. 47, No. 1-2 | 1 Mar 1986 Cross Ref A Note on functions with interlacing rootsJournal of Approximation Theory, Vol. 43, No. 1 | 1 Jan 1985 Cross Ref Optimal interpolation with incomplete polynomialsJournal of Approximation Theory, Vol. 41, No. 3 | 1 Jul 1984 Cross Ref The signal subspace approach for multiple wide-band emitter locationIEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 31, No. 6 | 1 Dec 1983 Cross Ref On the Polynomial and Rational Projections in the Complex PlaneL. BrutmanSIAM Journal on Numerical Analysis, Vol. 17, No. 3 | 17 July 2006AbstractPDF (712 KB)An efficient blind identification algorithm for multichannel FIR systems using linear predictionProceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181) Cross Ref Volume 17, Issue 3| 1980SIAM Journal on Numerical Analysis333-477 History Submitted:10 October 1979Published online:17 July 2006 InformationCopyright © 1980 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0717031Article page range:pp. 373-375ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics