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 $\text{O}(N)$ 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：我们针对多段不连续区间上正交多项式（目前尚无解析公式）开发了一种数值计算方法。该方法利用正交多项式的Fokas–Its–Kitaev Riemann–Hilbert表示，生成了一个$\text{O}(N)$算法来计算前$N$个递推系数。该方法还能在复平面内对多项式及其柯西变换进行逐点评估。通过采用切比雪夫多项式的加权柯西积分对权函数的奇异性进行编码，该方法的效率得到显著提升，优于现有其他技术。我们验证了该方法具有快速收敛性，并展示了其在可积系统与逼近理论中的应用。