Complex interpolating polynomials

Varma, A. K.

doi:10.1090/S0002-9939-1988-0938655-X

Complex interpolating polynomials
HTML articles powered by AMS MathViewer

by A. K. Varma

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

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

PDF | Request permission

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 $f \in C\left [ {|z| \leq 1} \right ]$. 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].

References

A. S. Cavaretta Jr., A. Sharma, and R. S. Varga, Hermite-Birkhoff interpolation in the $n$th roots of unity, Trans. Amer. Math. Soc. 259 (1980), no. 2, 621–628. MR 567101, DOI 10.1090/S0002-9947-1980-0567101-0
O. Kiš, Notes on interpolation, Acta Math. Acad. Sci. Hungar. 11 (1960), 49–64 (Russian). MR 113070, DOI 10.1007/BF02020623
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
S. D. Riemenschneider and A. Sharma, Birkhoff interpolation at the $n$th roots of unity: convergence, Canadian J. Math. 33 (1981), no. 2, 362–371. MR 617626, DOI 10.4153/CJM-1981-030-9
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, DOI 10.1137/0514061
J. Szabados, On some convergent interpolatory polynomials, Fourier analysis and approximation theory (Proc. Colloq., Budapest, 1976) Colloq. Math. Soc. János Bolyai, vol. 19, North-Holland, Amsterdam-New York, 1978, pp. 805–815. MR 540356
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
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, DOI 10.1016/0021-9045(80)90138-0
A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587

Convergence rates for some lacunary interpolators on their roots of unity