Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles

Анотація

For $a\in (0,1)$ let $L^k_m(a)$ be an error of the best approximation of the function $\mathrm{sgn}(x)$ on 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
$$\lim_{m\to\infty}L^k_m(a)\left(\displaystyle\frac{1+a}{1-a}\right)^{m-\frac{1}{2}}(2m-1)^{k+\frac{1}{2}}=\frac{2}{\pi}\left(\frac{1-a^2}{2a}\right)^{k+\frac{1}{2}}\Gamma\left(k+\frac{1}{2}\right).$$ 

Mathematics Subject Classification: 41A44, 30E.