Complex interpolating polynomials
Author:
A. K. Varma
Journal:
Proc. Amer. Math. Soc. 103 (1988), 125130
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].
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A. Sharma and J. Szabados, Convergence rates for some lacunary interpolators on their roots of unity (preprint April 1987).
 [1]
 A. S. Cavaretta, A. Sharma and R. S. Varga, Hermite Birkhoff interpolation in the th roots of unity, Trans. Amer. Math. Soc. 259 (1980), 621628. MR 567101 (81c:30064)
 [2]
 O. Kis, Notes on interpolation, Acta Math. Acad. Sci. Hungar. 11 (1960), 4964. 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 th roots of unity, Canad. J. Math. 33 (1981), 362371. MR 617626 (83a:41005)
 [5]
 A. Sharma and P. Vertesi, Mean convergence of interpolation in roots of unity, SIAM J. Math. Anal. 14 (1983), 800806. 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, NorthHolland, 1978, pp. 805816. 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), 255264. MR 847546 (87j:41013)
 [8]
 P. Turán, On some open problems of approximation theory, J. Approximation Theory 29 (1980), 2385. 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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DOI:
http://dx.doi.org/10.1090/S0002993919880938655X
PII:
S 00029939(1988)0938655X
Article copyright:
© Copyright 1988
American Mathematical Society
