Geodesic
Rational Approximation to x n II
Canadian journal of mathematics, Tome 32 (1980) no. 2, pp. 310-316

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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 + 1σx n 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, a 1, a 2, a 3, ..., an be any given real numbers. Then for every N > n, where λ is numerically the smallest root of the polynomial with 
Newman, D. J.; Reddy, A. R. Rational Approximation to x n II. Canadian journal of mathematics, Tome 32 (1980) no. 2, pp. 310-316. doi: 10.4153/CJM-1980-023-9
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[1] 1. Achieser, N. I., Theory of approximation (Frederick Ungar Publishing Co., New York, 1956). Google Scholar

[2] 2. Newman, D. J., Rational approximation to xn, J. Approximation Theor. 22 (1978), 285–288. Google Scholar

[3] 3. Newman, D. J. and Reddy, A. R., Rational approximation to xn, Pacific J. Mathematic. 67 (1976), 247–250. Google Scholar

[4] 4. Timan, A. F., Theory of approximation of functions of a real variable, International series of monographs in pure and applied mathematics 34 (Macmillan, New York, 1963). Google Scholar

[5] 5. Voronovskaja, E. V., The functional method and its applications (English translation), American Math. Society, Providence, RI (1970). Google Scholar

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