We provide explicit asymptotically optimal quadrature rules for uniform C^k-splines, k = 0,1, over the real line. The nodes of these quadrature rules are given in terms of the zeros of ultraspherical polynomials (Gegenbauer polynomials) and related polynomials. We conjecture that our derived rules are the only possible periodic asymptotically optimal quadrature rules for these spline spaces.

中文翻译：

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实线上均匀样条的渐近最优正交规则

我们为实线上的均匀C ^k-样条线k = 0,1提供了明确的渐近最优正交规则。这些正交规则的节点以超球面多项式（Gegenbauer多项式）和相关多项式的零值给出。我们推测，对于这些样条空间，我们的导出规则是唯一可能的周期渐近最优正交规则。