Previous works show convergence of rational Chebyshev approximants to the Pad\'e approximant as the underlying domain of approximation shrinks to the origin. In the present work, the asymptotic error and interpolation properties of rational Chebyshev approximants are studied in such settings. Namely, the point-wise error of Chebyshev approximants is shown to approach a Chebyshev polynomial multiplied by the asymptotically leading order term of the error of the Pad\'e approximant, and similar results hold true for the uniform error and Chebyshev constants. Moreover, rational Chebyshev approximants are shown to attain interpolation nodes which approach scaled Chebyshev nodes in the limit. Main results are formulated for interpolatory best approximations and apply for complex Chebyshev approximation as well as real Chebyshev approximation to real functions and unitary best approximation to the exponential function.

翻译：先前的研究表明，当逼近的底层域收缩至原点时，有理切比雪夫逼近式收敛于帕德逼近式。本文在此类设定下研究了有理切比雪夫逼近式的渐近误差与插值性质。具体而言，切比雪夫逼近式的逐点误差被证明趋近于一个切比雪夫多项式乘以帕德逼近式误差的渐近主导项，且类似结论对一致误差与切比雪夫常数同样成立。此外，有理切比雪夫逼近式被证明可获得在极限下趋近于缩放切比雪夫节点的插值节点。主要结论针对插值最佳逼近建立，并适用于复切比雪夫逼近、实函数实切比雪夫逼近以及指数函数的酉最佳逼近。