Uniform approximation of sgn(x) by rational functions with prescribed poles

2006/08/10 by Peherstorfer, Franz, Yuditskii, Peter
#30E #41A44 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.math/0608253

Abstract

For a∈ (0,1) let L^k_m(a) be the error of the best approximation of the function \sgn(x) on the two symmetric intervals [-1,-a]∪[a,1] by rational functions with the only possible poles of degree 2k-1 at the origin and of 2m-1 at infinity. Then the following limit exists lim_{m→ ∞}L^k_m(a)((1+a)/(1-a))^m-1/2 (2m-1)^k+1/2=\frac 2 π((1-a²)/(2a))^k+1/2 Γ(k+\frac 1 2).

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