In this paper, new and optimal asymptotics on the decay of the coefficients for functions of limited regularity expanded in terms of Jacobi and Gegenbauer polynomial series are presented. For a class of functions with interior singularities, the decay of the coefficient is of the same asymptotic order for arbitrary $$\alpha ,\,\beta >-1$$ α , β > - 1 , which confirms that the decay of the coefficients in the Jacobi polynomial series without normalization is a factor of $$ \sqrt{n}$$ n slower compared with the Chebyshev expansion. While for functions with boundary singularities, the decay depends on $$\alpha $$ α and $$\beta $$ β with $$\alpha ,\,\beta >-1$$ α , β > - 1 . For Gegenbauer expansion, it is related to the parameter $$\lambda $$ λ whatever f with interior or boundary singularities. All of these asymptotic analysis are optimal. Moreover, under the optimal asymptotic analysis, it derives that the truncated spectral expansions with some specific parameters can achieve the optimal convergence rates, i.e., the same as the best polynomial approximation in the sense of absolute maximum error norm. Numerical examples illustrate the perfect coincidence with the estimates.

中文翻译：

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有限正则函数的正交多项式展开渐近线的最优衰减率

在本文中，提出了根据 Jacobi 和 Gegenbauer 多项式级数展开的有限正则函数的系数衰减的新的和最优的渐近性。对于一类具有内部奇点的函数，系数的衰减对于任意 $$\alpha ,\,\beta >-1$$ α , β > - 1 具有相同的渐近阶，这证实了与 Chebyshev 展开相比，没有归一化的 Jacobi 多项式级数中的系数是 $$\sqrt{n}$$n 的一个因子。而对于具有边界奇点的函数，衰减取决于 $$\alpha $$ α 和 $$\beta $$ β ，其中 $$\alpha ,\,\beta >-1$$ α , β > - 1 。对于 Gegenbauer 展开，无论 f 具有内部奇点还是边界奇点，它都与参数 $$\lambda $$ λ 相关。所有这些渐近分析都是最优的。此外，在最优渐近分析下，推导出具有某些特定参数的截断谱展开可以达到最优收敛速度，即与绝对最大误差范数意义上的最佳多项式近似相同。数值例子说明了与估计的完美重合。