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

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

2006-08-10

arxiv.org/abs/math/0608253v1

Mathematics

Classical Analysis and ODEs

Scientific paper

For $a\in (0,1)$ let $L^k_m(a)$ be the error of the best approximation of the function $\sgn(x)$ on the two symmetric intervals $[-1,-a]\cup[a,1]$ 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 \begin{equation} \lim_{m\to \infty}L^k_m(a)(\frac{1+a}{1-a})^{m-{1/2}} (2m-1)^{k+{1/2}}=\frac 2 \pi(\frac{1-a^2}{2a})^{k+{1/2}} \Gamma(k+\frac 1 2). \end{equation}

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