Complex interpolating polynomials

Author:
A. K. Varma

Journal:
Proc. Amer. Math. Soc. **103** (1988), 125-130

MSC:
Primary 30E05; Secondary 41A05

MathSciNet review:
938655

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Abstract: Let be the unique interpolatory polynomial of degree satisfying the conditions given by (1.1) where the 's are the zeros of the polynomial . The object of this paper is to consider the rate of convergence of to in the norm where . This problem was initially raised by P. Turán in the case and in this case the solution was obtained by J. Szabados and A. K. Varma in [**7**].

**[1]**A. S. Cavaretta Jr., A. Sharma, and R. S. Varga,*Hermite-Birkhoff interpolation in the 𝑛th roots of unity*, Trans. Amer. Math. Soc.**259**(1980), no. 2, 621–628. MR**567101**, 10.1090/S0002-9947-1980-0567101-0**[2]**O. Kiš,*Notes on interpolation*, Acta Math. Acad. Sci. Hungar**11**(1960), 49–64 (Russian). MR**0113070****[3]**George G. Lorentz, Kurt Jetter, and Sherman D. Riemenschneider,*Birkhoff interpolation*, Encyclopedia of Mathematics and its Applications, vol. 19, Addison-Wesley Publishing Co., Reading, Mass., 1983. MR**680938****[4]**S. D. Riemenschneider and A. Sharma,*Birkhoff interpolation at the 𝑛th roots of unity: convergence*, Canad. J. Math.**33**(1981), no. 2, 362–371. MR**617626**, 10.4153/CJM-1981-030-9**[5]**A. Sharma and P. Vértesi,*Mean convergence and interpolation in roots of unity*, SIAM J. Math. Anal.**14**(1983), no. 4, 800–806. MR**704493**, 10.1137/0514061**[6]**J. Szabados,*On some convergent interpolatory polynomials*, Fourier analysis and approximation theory (Proc. Colloq., Budapest, 1976), Vol. II, Colloq. Math. Soc. János Bolyai, vol. 19, North-Holland, Amsterdam-New York, 1978, pp. 805–815. MR**540356****[7]**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**, 10.1016/0021-9045(86)90034-1**[8]**P. Turán,*On some open problems of approximation theory*, J. Approx. Theory**29**(1980), no. 1, 23–85. P. Turán memorial volume; Translated from the Hungarian by P. Szüsz. MR**595512**, 10.1016/0021-9045(80)90138-0**[9]**A. Zygmund,*Trigonometric series: Vols. I, II*, Second edition, reprinted with corrections and some additions, Cambridge University Press, London-New York, 1968. MR**0236587****[10]**A. Sharma and J. Szabados,*Convergence rates for some lacunary interpolators on their roots of unity*(preprint April 1987).

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DOI:
https://doi.org/10.1090/S0002-9939-1988-0938655-X

Article copyright:
© Copyright 1988
American Mathematical Society