A unified theory for real vs. complex rational Chebyshev approximation on an interval

Ruttan, Arden; Varga, Richard S.

doi:10.1090/S0002-9947-1989-0948196-7

A unified theory for real vs. complex rational Chebyshev approximation on an interval
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by Arden Ruttan and Richard S. Varga PDF

Trans. Amer. Math. Soc. 312 (1989), 681-697 Request permission

Abstract:

A unified approach is presented for determining all the constants ${\gamma _{m,n}}\;(m \geq 0,n \geq 0)$ which occur in the study of real vs. complex rational Chebyshev approximation on an interval. In particular, it is shown that ${\gamma _{m,m + 2}} = 1/3\;(m \geq 0)$, a problem which had remained open.

References

A. L. Levin, On the degree of complex rational approximation to real functions, Constr. Approx. 2 (1986), no. 3, 213–219. MR 891971, DOI 10.1007/BF01893427
Günter Meinardus, Approximation of functions: Theory and numerical methods, Expanded translation of the German edition, Springer Tracts in Natural Philosophy, Vol. 13, Springer-Verlag New York, Inc., New York, 1967. Translated by Larry L. Schumaker. MR 0217482
Arden Ruttan and Richard S. Varga, Real vs. complex rational Chebyshev approximation on an interval, Rocky Mountain J. Math. 19 (1989), no. 1, 375–381. Constructive Function Theory—86 Conference (Edmonton, AB, 1986). MR 1016188, DOI 10.1216/RMJ-1989-19-1-375
E. B. Saff and R. S. Varga, Nonuniqueness of best complex rational approximations to real functions on real intervals, J. Approx. Theory 23 (1978), no. 1, 78–85. MR 499031, DOI 10.1016/0021-9045(78)90081-3
Lloyd N. Trefethen and Martin H. Gutknecht, Real vs. complex rational Chebyshev approximation on an interval, Trans. Amer. Math. Soc. 280 (1983), no. 2, 555–561. MR 716837, DOI 10.1090/S0002-9947-1983-0716837-X
Richard S. Varga, Topics in polynomial and rational interpolation and approximation, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 81, Presses de l’Université de Montréal, Montreal, Que., 1982. MR 654329