On Hermite-Fejér interpolation in a Jordan domain
HTML articles powered by AMS MathViewer
- by Charles K. Chui and Xie Chang Shen PDF
- Trans. Amer. Math. Soc. 323 (1991), 93-109 Request permission
Abstract:
The Hermite-Fejér interpolation problem on a Jordan domain is studied. Under certain mild conditions on the smoothness of the boundary curve, we give both uniform and ${L^p}$, $0 < p < \infty$, estimates on the rate of convergence. Our estimates are sharp even for the unit disk setting.References
-
S. Ya. Al’per, On the approximation in the mean to analytic functions in classes ${E_p}$, Investigation of Modern Problems of Theory of Functions of a Complex Variable, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1960, pp. 273-286.
- S. Ja. Al′per and G. I. Kalinogorskaja, The convergence of Lagrange interpolation polynomials in the complex domain, Izv. Vysš. Učebn. Zaved. Matematika 1969 (1969), no. 11 (90), 13–23 (Russian). MR 0259138
- J. H. Curtiss, Convergence of complex Lagrange interpolation polynomials on the locus of the interpolation points, Duke Math. J. 32 (1965), 187–204. MR 210902, DOI 10.1215/S0012-7094-65-03217-5
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
- V. K. Dzyadyk, Vvedenie v teoriyu ravnomernogo priblizheniya funktsiĭ polinomami, Izdat. “Nauka”, Moscow, 1977 (Russian). MR 0612836
- Dieter Gaier, Über Interpolation in regelmässig verteilten Punkten mit Nebenbedingungen, Math. Z. 61 (1954), 119–133 (German). MR 68004, DOI 10.1007/BF01181337
- Dieter Gaier, Vorlesungen über Approximation im Komplexen, Birkhäuser Verlag, Basel-Boston, Mass., 1980 (German). MR 604011, DOI 10.1007/978-3-0348-5812-0 Xie-Chang Shen, On the order of approximation by Hermite-Fejér polynomials in a Jordan domain, Approximation Theory VI, Academic Press, New York, 1989. —, The convergence of $(0,1,\ldots ,q)$ Hermite-Fejér interpolating polynomials on the roots of unity, Chinese Ann. Math. (to appear).
- Xie Chang Shen and Le Fan Zhong, The mean order of approximation by Lagrange interpolation polynomials in the complex plane, Kexue Tongbao (Chinese) 33 (1988), no. 11, 810–814 (Chinese). MR 964197
- J. Szabados and A. K. Varma, On an open problem of P. Turán concerning Birkhoff interpolation based on the roots of unity, J. Approx. Theory 47 (1986), no. 3, 255–264. MR 847546, DOI 10.1016/0021-9045(86)90034-1
- A. K. Varma, Complex interpolating polynomials, Proc. Amer. Math. Soc. 103 (1988), no. 1, 125–130. MR 938655, DOI 10.1090/S0002-9939-1988-0938655-X
- P. Vértesi, Linear operators on the roots of unity, Studia Sci. Math. Hungar. 15 (1980), no. 1-3, 241–245. MR 681444 C. M. Wu, The boundary properties of conformal mapping, Acta Math. Sinica 7 (1957), 271-276. A. Zygmund, Trigonometric series, Cambridge Univ. Press, Cambridge, 1968.
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 323 (1991), 93-109
- MSC: Primary 41A05; Secondary 41A10, 41A25
- DOI: https://doi.org/10.1090/S0002-9947-1991-1018573-6
- MathSciNet review: 1018573