标题： 切比雪夫逼近在收缩域上的误差 显示英文标题

标题： The error of Chebyshev approximations on shrinking domains

摘要： 先前的研究表明，随着逼近的基域缩小到原点，有理切比雪夫逼近子收敛于帕德逼近子。在本工作中，研究了这种情况下有理切比雪夫逼近子的渐近误差和插值性质。具体来说，显示了切比雪夫逼近子的逐点误差趋于一个切比雪夫多项式乘以帕德逼近子误差的渐近主导项，对于一致误差和切比雪夫常数也有类似的结果。此外，显示有理切比雪夫逼近子达到的插值节点在极限情况下接近缩放后的切比雪夫节点。主要结果适用于插值最佳逼近，并适用于复数切比雪夫逼近以及实数切比雪夫逼近实函数和指数函数的单位最佳逼近。 显示英文摘要

摘要： 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.