Berriochoa, E.; Cachafeiro, A.; Brey, E. Martínez Some improvements to the Hermite-Fejér interpolation on the circle and bounded interval. (English) Zbl 1217.65028 Comput. Math. Appl. 61, No. 4, 1228-1240 (2011). Summary: We study the convergence of the Hermite-Fejér and the Hermite interpolation polynomials, which are constructed by taking equally spaced nodes on the unit circle. The results that we obtain are concerned with the behaviour outside and inside the unit circle, when we consider analytic functions on a suitable domain. As a consequence, we achieve some improvements on Hermite interpolation problems on the real line. Since the Hermite-Fejér and the Hermite interpolation problems on \([-1,1]\), with nodal systems mainly based on sets of zeros of orthogonal polynomials, have been widely studied, in our contribution we develop a theory for three special nodal systems. They are constituted by the zeros of the Tchebychef polynomial of the second kind joint with the extremal points \(- 1\) and \(1\), the zeros of the Tchebychef polynomial of the fourth kind joint with the point \(-1\), and the zeros of the Tchebychef polynomial of the third kind joint with the point \(1\). We present a simple and efficient method to compute these interpolation polynomials and we study the convergence properties. Cited in 8 Documents MSC: 65D05 Numerical interpolation 41A05 Interpolation in approximation theory Keywords:Hermite interpolation; Hermite-Fejér interpolation; Laurent polynomials; convergence; unit circle PDFBibTeX XMLCite \textit{E. Berriochoa} et al., Comput. Math. Appl. 61, No. 4, 1228--1240 (2011; Zbl 1217.65028) Full Text: DOI References: [1] Szabados, J., On Hermite-Fejér interpolation for the Jacobi abscissa, Acta Math. Acad. Sci. Hungar., 23, 449-464 (1972) · Zbl 0253.41004 [2] Vértesi, P., Notes on the Hermite-Fejér interpolation based on the Jacobi abscissas, Acta Math. Acad. Sci. Hungar., 24, 233-239 (1973) · Zbl 0267.41001 [3] Daruis, L.; González-Vera, P., A note on Hermite-Fejér interpolation for the unit circle, Appl. Math. Lett., 14, 997-1003 (2001) · Zbl 0982.41003 [4] Berriochoa, E.; Cachafeiro, A., Algorithms for solving Hermite interpolation problems using the fast Fourier transform, J. Comput. Appl. Math., 235, 882-894 (2010) · Zbl 1215.33009 [5] Brutman, L.; Gopengauz, I., On divergence of Hermite-Fejér interpolation to \(f(z) = z\) in the complex plane, Constr. Approx., 15, 611-617 (1999) · Zbl 0939.41005 [6] Brutman, L.; Gopengauz, I.; Vértesi, P., On the domain of divergence of Hermite-Fejér interpolating polynomials, J. Approx. Theory, 106, 287-290 (2000) · Zbl 0970.41001 [7] Davis, P. J., Interpolation and Approximation (1975), Dover Publications: Dover Publications New York · Zbl 0111.06003 [8] Rivlin, T., The Chebyshev polynomials, (Pure and Applied Mathematics (1974), John Wiley & Sons: John Wiley & Sons New York) · Zbl 0299.41015 [9] Szegő, G., (Orthogonal Polynomials. Orthogonal Polynomials, Amer. Math. Soc. Coll. Publ., vol. 23 (1975), Amer. Math. Soc.: Amer. Math. Soc. Providence) [10] J.M. García Amor, Ortogonalidad Bernstein-Chebyshev en la recta real, Doctoral Dissertation, Universidad de Vigo (2003) (in Spanish).; J.M. García Amor, Ortogonalidad Bernstein-Chebyshev en la recta real, Doctoral Dissertation, Universidad de Vigo (2003) (in Spanish). [11] Criscuolo, G.; Della Vecchia, B.; Mastroianni, G., Approximation by extended Hermite-Féjer and Hermite interpolation, (Colloquia Mathematica Societatis J. Bolyai, vol. 58 (1990), North Holland) · Zbl 0768.41003 [12] Szabados, J.; Vértesi, P., Interpolation of Functions (1990), World Scientific: World Scientific Singapore · Zbl 0721.41003 [13] Szász, P., On a sum concerning the zeros of the Jacobi polynomials with application to the theory of generalized quasi-step parabolas, Monatsh. Math., 68, 167-174 (1964) · Zbl 0128.06502 [14] Szász, P., The extended Hermite-Fejér interpolation formula with application to the theory of generalized almost-step parabolas, Publ. Math. Debrecen, 11, 85-100 (1964) · Zbl 0154.05901 [15] Kincaid, D.; Cheney, W., Numerical Analysis: Mathematics of Scientific Computation (1991), Brooks/Cole Publishing Company [16] Stoer, J.; Bulirsch, R., Introduction to Numerical Analysis (1996), Springer: Springer New York · Zbl 1004.65001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.