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Rational Approximation to xn II

Published online by Cambridge University Press:  20 November 2018

D. J. Newman  and
A. R. Reddy
D. J. Newman
Affiliation:
Temple University, Philadelphia, Pennsylvania
A. R. Reddy
Affiliation:
University of Hyderabad, Hyderabad, India
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Introduction. In 1858 Chebyshev showed that xn+l can be approximated uniformly on [–1, 1] by polynomials of degree at most n with an error 2–n. Let 0 ≦ σ ≦ (n + l)tan2(π/2n + 2). In 1868 Zolotarev established that x n + 1σxn can be approximated uniformly on [ –1, 1] by polynomials of degree at most (n – 1) with an error 2–n(l + σ/n + l)n+1. It is interesting to note that for the case σ = 0, Zolotarev's result includes Chebyshev's result. Achieser ([1], p. 279) proved the following analogue for rational approximation. Let a0 ≠ 0, a1, a2, a3, …, an be any given real numbers. Then for every N > n,

where λ is numerically the smallest root of the polynomial

with

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 2 , February 1980 , pp. 310 - 316
Copyright
Copyright © Canadian Mathematical Society 1980

References

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