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Complex interpolating polynomials


Author: A. K. Varma
Journal: Proc. Amer. Math. Soc. 103 (1988), 125-130
MSC: Primary 30E05; Secondary 41A05
DOI: https://doi.org/10.1090/S0002-9939-1988-0938655-X
MathSciNet review: 938655
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Abstract: Let $ {I_{n,m}}\left( {f,z} \right)$ be the unique interpolatory polynomial of degree $ \leq 2n - 1$ satisfying the conditions given by (1.1) where the $ {z_{kn}}$'s are the zeros of the polynomial $ {z^n} - 1$. The object of this paper is to consider the rate of convergence of $ {I_{m,n}}\left( {f,z} \right)$ to $ f\left( z \right)$ in the $ {L_p}$ norm where $ f \in C\left[ {\vert z\vert \leq 1} \right]$. This problem was initially raised by P. Turán in the case $ p = 2$ and in this case the solution was obtained by J. Szabados and A. K. Varma in [7].


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  • [1] A. S. Cavaretta, A. Sharma and R. S. Varga, Hermite Birkhoff interpolation in the $ n$th roots of unity, Trans. Amer. Math. Soc. 259 (1980), 621-628. MR 567101 (81c:30064)
  • [2] O. Kis, Notes on interpolation, Acta Math. Acad. Sci. Hungar. 11 (1960), 49-64. MR 0113070 (22:3911)
  • [3] G. G. Lorentz, K. Jetter and S. D. Riemenschneider, Birkhoff interpolation, Encyclopedia of Mathematics and its Application, Vol. 19, (see Chapter XII), 1983. MR 680938 (84g:41002)
  • [4] S. D. Riemenschneider and A. Sharma, Birkhoff interpolation at the $ n$th roots of unity, Canad. J. Math. 33 (1981), 362-371. MR 617626 (83a:41005)
  • [5] A. Sharma and P. Vertesi, Mean convergence of interpolation in roots of unity, SIAM J. Math. Anal. 14 (1983), 800-806. MR 704493 (85h:30051)
  • [6] J. Szabados, On some convergent interpolatory polynomials, Fourier Analysis and Approximation Theory (G. Alexits and P. Turan, eds.), vol. II, North-Holland, 1978, pp. 805-816. MR 540356 (81g:41009)
  • [7] J. Szabados and A. K. Varma, On an open problem of P. Turan concerning Birkhoff interpolation based on the roots of unity, J. Approximation Theory 47 (1986), 255-264. MR 847546 (87j:41013)
  • [8] P. Turán, On some open problems of approximation theory, J. Approximation Theory 29 (1980), 23-85. MR 595512 (82e:41003)
  • [9] A. Zygmund, Trigonometric series, vols. I, II, Cambridge Univ. Press, 1968. MR 0236587 (38:4882)
  • [10] A. Sharma and J. Szabados, Convergence rates for some lacunary interpolators on their roots of unity (preprint April 1987).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0938655-X
Article copyright: © Copyright 1988 American Mathematical Society

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