The aim of this article is to present the essential properties of a finite class of orthogonal polynomials related to the probability density function of the F‐distribution over the positive real line. We introduce some basic properties of the Romanovski–Jacobi polynomials, the Romanovski–Jacobi–Gauss type quadrature formulae and the associated interpolation, discrete transforms, spectral differentiation and integration techniques in the physical and frequency spaces, and basic approximation results for the weighted projection operator in the nonuniformly weighted Sobolev space. We discuss the relationship between such kinds of finite orthogonal polynomials and other classes of infinite orthogonal polynomials. Moreover, we derive spectral Galerkin schemes based on a Romanovski–Jacobi expansion in space and time to solve the Cauchy problem for a scalar linear hyperbolic equation in one and two space dimensions posed in the positive real line. Two numerical examples demonstrate the robustness and accuracy of the schemes.

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关于Romanovski–Jacobi多项式及其相关的逼近结果

本文的目的是介绍与F的概率密度函数有关的有限类正交多项式的基本性质在正实线上分布。我们介绍了Romanovski–Jacobi多项式的一些基本性质，Romanovski–Jacobi–Gauss型正交公式以及相关的插值，离散变换，物理和频率空间中的离散变换，频谱微分和积分技术，以及加权投影算符的基本近似结果在非均匀加权Sobolev空间中 我们讨论了这类有限正交多项式与其他类别的无限正交多项式之间的关系。此外，我们基于空间和时间上的罗曼诺夫斯基-雅各比展开推导频谱加勒金方案，以解决正实线中一维和二维空间中标量线性双曲方程的柯西问题。