# Asymptotic distribution of poles and zeros of best rational approximants to ${x}^{\alpha}$ on [0,1]

Banach Center Publications (1995)

- Volume: 31, Issue: 1, page 329-348
- ISSN: 0137-6934

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topSaff, E., and Stahl, H.. "Asymptotic distribution of poles and zeros of best rational approximants to $x^α$ on [0,1]." Banach Center Publications 31.1 (1995): 329-348. <http://eudml.org/doc/262780>.

@article{Saff1995,

abstract = {Let $r_n* ∈ ℛ_\{nn\}$ be the best rational approximant to $f(x) = x^α$, 1 > α > 0, on [0,1] in the uniform norm. It is well known that all poles and zeros of $r_n*$ lie on the negative axis $ℝ_\{<0\}$. In the present paper we investigate the asymptotic distribution of these poles and zeros as n → ∞. In addition we determine the asymptotic distribution of the extreme points of the error function $e_n = f - r_n*$ on [0,1], and survey related convergence results.},

author = {Saff, E., Stahl, H.},

journal = {Banach Center Publications},

keywords = {rational approximation; best approximation; distribution of poles and zeros},

language = {eng},

number = {1},

pages = {329-348},

title = {Asymptotic distribution of poles and zeros of best rational approximants to $x^α$ on [0,1]},

url = {http://eudml.org/doc/262780},

volume = {31},

year = {1995},

}

TY - JOUR

AU - Saff, E.

AU - Stahl, H.

TI - Asymptotic distribution of poles and zeros of best rational approximants to $x^α$ on [0,1]

JO - Banach Center Publications

PY - 1995

VL - 31

IS - 1

SP - 329

EP - 348

AB - Let $r_n* ∈ ℛ_{nn}$ be the best rational approximant to $f(x) = x^α$, 1 > α > 0, on [0,1] in the uniform norm. It is well known that all poles and zeros of $r_n*$ lie on the negative axis $ℝ_{<0}$. In the present paper we investigate the asymptotic distribution of these poles and zeros as n → ∞. In addition we determine the asymptotic distribution of the extreme points of the error function $e_n = f - r_n*$ on [0,1], and survey related convergence results.

LA - eng

KW - rational approximation; best approximation; distribution of poles and zeros

UR - http://eudml.org/doc/262780

ER -

## References

top- [Be1] S. Bernstein, Sur meilleure approximation de |x| par des polynômes de degrés donnés, Acta Math. 37 (1913), 1-57. Zbl44.0475.01
- [Be2] S. Bernstein, About the best approximation of ${\left|x\right|}^{p}$ by means of polynomials of very high degree, Bull. Acad. Sci. USRR, Cl. Math. Nat. Sci. 2 (1938), 169-190; also in: Collected Works, Vol. II, 262-272 (in Russian).
- [Bu1] A. P. Bulanov, Asymptotics for the least deviation of |x| from rational functions, Math. Sb. 76 (118) (1968), 288-303 (in Russian); English transl.: Math. USSR-Sb. 5 (1968), 275-290.
- [Bu2] A. P. Bulanov, The approximation of ${x}^{1/3}$ by rational functions, Vesci Akad. Navuk BSSR Ser. Fiz.-Navuk 1968, no. 2, 47-56 (in Russian).
- [FrSz] G. Freud and J. Szabados, Rational Approximation to ${x}^{\alpha}$, Acta Math. Acad. Sci. Hungar. 18 (1967), 393-399. Zbl0172.34201
- [Ga] T. Ganelius, Rational approximation of ${x}^{\alpha}$ on [0,1], Anal. Math. 5 (1979), 19-33. Zbl0425.41017
- [Go1] A. A. Gonchar, On the speed of rational approximation of continuous functions with characteristic singularities, Math. Sb. 73 (115) (1967), 630-638 (in Russian); English transl.: Math. USSR-Sb. 2 (1967). Zbl0175.06001
- [Go2] A. A. Gonchar, Rational approximation of the function ${x}^{\alpha}$, in: Constructive Theory of Functions (Proc. Internat. Conf., Varna 1970), Izdat. Bolgar. Akad. Nauk, Sofia, 1972, 51-53 (in Russian).
- [Go3] A. A. Gonchar, The rate of rational approximation and the property of single valuedness of an analytic function in a neighborhood of an isolated singular point, Mat. Sb. 94 (136) (1974), 265-282 (in Russian); English transl.: Math. USSR-Sb. 23 (1974).
- [Me] G. Meinardus, Approximation of Functions: Theory and Numerical Methods, Springer, New York 1967.
- [Ne] D. J. Newman, Rational approximation to |x|, Michigan Math. J. 11 (1964), 11-14.
- [Ri] T. J. Rivlin, An Introduction to the Approximation of Functions, Blaisdell, Waltham, Mass., 1969.
- [SaSt1] E. B. Saff and H. Stahl, Distribution of poles and zeros of best approximations for ${\left|x\right|}^{\alpha}$, in: Constructive Theory of Functions 91, G. K. Ivanov et al. (eds.), Publ. of the Bulgarian Acad., Sofia, 1992, 249-257.
- [SaSt2] E. B. Saff and H. Stahl, Sequences in the Walsh table for ${\left|x\right|}^{\alpha}$, J. Canad. Math. Soc., to appear.
- [St1] H. Stahl, Best uniform rational approximation of |x| on [-1,1], Mat. Sb. 183 (1992), 85-112 (in Russian); English transl.: Russian Acad. Sci. Sb. Math. 76 (1993), 461-487.
- [St2] H. Stahl, Uniform approximation of |x|, in: Methods of Approximation Theory in Complex Analysis and Mathematical Physics, A. A. Gonchar and E. B. Saff (eds.), Nauka, 1992, 110-130.
- [St3] H. Stahl, Best uniform rational approximation of ${x}^{\alpha}$ on [0,1], Bull. Amer. Math. Soc. 28 (1993), 116-122.
- [St4] H. Stahl, Poles and zeros of best rational approximations of |x|, Constr. Approx. 10 (1994), 469-522. Zbl0815.41015
- [StTo] H. Stahl and V. Totik, General Orthogonal Polynomials, Encyclopedia of Math. Appl. 43, Cambridge University Press, Cambridge, 1992.
- [VC] R. S. Varga and A. J. Carpenter, Some numeral results on best uniform rational approximation of ${x}^{\alpha}$ on [0,1], Numer. Algorithms 2 (1992), 171-185.
- [Vy1] N. S. Vyacheslavov, The approximation of |x| by rational functions, Mat. Zametki 16 (1974), 163-171 (in Russian). Zbl0307.41010
- [Vy2] N. S. Vyacheslavov, On the approximation of |x| by rational functions, Dokl. Akad. Nauk SSSR 220 (1975), 512-515; English transl.: Soviet Math. Dokl. 16 (1975), 100-104.
- [Vy3] N. S. Vyacheslavov, On the approximation of ${x}^{\alpha}$ by rational functions, Izv. Akad. Nauk SSSR 44 (1980) (in Russian); English transl.: Math. USSR-Izv. 16 (1981), 83-101.

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