We develop a numerical method for computing with orthogonal polynomials that are orthogonal on multiple, disjoint intervals for which analytical formulae are currently unknown. Our approach exploits the Fokas--Its--Kitaev Riemann--Hilbert representation of the orthogonal polynomials to produce an $\OO(N)$ method to compute the first $N$ recurrence coefficients. The method can also be used for pointwise evaluation of the polynomials and their Cauchy transforms throughout the complex plane. The method encodes the singularity behavior of weight functions using weighted Cauchy integrals of Chebyshev polynomials. This greatly improves the efficiency of the method, outperforming other available techniques. We demonstrate the fast convergence of our method and present applications to integrable systems and approximation theory.
翻译:我们开发了一种数字方法, 用于与正对齐的多面体进行计算, 这些多面体在多个断开的间隔上正对对齐, 目前还不知道分析公式。 我们的方法利用正对齐面体的Fokas- Its- Kitaev Riemann- Hilbert 表示, 以生成一个$\OO( N) 的方法来计算第一个 $N 重现系数。 该方法还可以用于对多面体及其在整个复杂平面上的突变进行有分辨的评估。 该方法用Chebyshev 多元面体的加权宽宽度组合来编码重量函数的奇比谢夫奇比谢夫单面体的奇比谢夫单面体特性。 这大大提高了该方法的效率, 超过了其他可用的技术。 我们展示了我们的方法的快速趋同, 并展示了对无法识别的系统和近似理论的应用 。</s>