An analogue of Montel’s theorem for some classes of rational functions

R. K. Kovacheva; Julian Lawrynowicz

Displaying similar documents to “An analogue of Montel’s theorem for some classes of rational functions”

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Rational interpolants with preassigned poles, theoretical aspects

Amiran Ambroladze, Hans Wallin (1999)

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Let ⨍ be an analytic function on a compact subset K of the complex plane ℂ, and let $r_{n} (z)$ denote the rational function of degree n with poles at the points ${b_{n i}}_{i = 1}^{n}$ and interpolating ⨍ at the points ${a_{n i}}_{i = 0}^{n}$ . We investigate how these points should be chosen to guarantee the convergence of $r_{n}$ to ⨍ as n → ∞ for all functions ⨍ analytic on K. When K has no “holes” (see [8] and [3]), it is possible to choose the poles ${b_{n i}}_{i, n}$ without limit points on K. In this paper we study the case of general compact sets K, when...

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

E. Saff, H. Stahl (1995)

Banach Center Publications

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Let $r_{n} * \in ℛ_{n n}$ 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.