On convergence of rational and best rational approximations
It is shown first that for a function f analytic everywhere except for the set of capacity zero and or arbitrary \(\eta >0\) (but not for \(\eta =0\), in general) interpolation at only \((1+\eta)\max\{M_ k,L_ k\}\) points (rather than the usual \(M_ k+L_ k+1\) points) is sufficient for a sequence of rational functions (of numerator degree \(\leq L_ k\), denominator degree \(\leq M_ k\), where \(1/\lambda\leq M_ k/L_ k\leq\lambda \) for some \(\lambda \geq 1\), and where \(M_ k,L_ k\to\infty )\) to converge in capacity as \(k\to\infty \), togethwer with all their derivatives. This impressive result, whose proof takes nearly six pages, generalizes previous results of Pommerenke and Wallin. It is applied to functions f that are additionally real on \({\mathbb{R}}\) and to their best real rational approximations on a series of real intervals. Further similar results concern the case where \(L_ k\to\infty \), \(M_ k/L_ k\to 0\). Some more recent related work is cited at the end of the paper.
scientific article; zbMATH DE number 3873672
scientific article