In this paper we mainly focus on the asymptotic expansion and quadrature rule for a class of singular-oscillatory-Bessel-type transforms. The asymptotic expansion in inverse powers of the frequency $\omega$ is obtained after avoiding the singularities. Then, based on the asymptotic expansion, we successfully construct a so-called modified Filon-type quadrature rule. Meanwhile, we also give the corresponding error analysis on two rules in inverse powers of the large frequency $\omega$. Some numerical examples based on theoretical results are presented to demonstrate the efficiency and accuracy of the proposed rules.

中文翻译：

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一类奇振荡贝塞尔型变换的渐近展开和正交规则

在本文中，我们主要关注一类奇异振荡Bessel型变换的渐近展开和正交规则。频率反幂的渐近展开$\omega$在避免奇异点之后获得。然后，基于渐近展开，我们成功构造了一个所谓的修正Filon型正交规则。同时，我们还针对大频率反幂的两个规则给出了相应的误差分析$\omega$。给出了一些基于理论结果的数值例子，以证明所提出规则的有效性和准确性。