关于乘法雅可比多项式与通过乘法级数进行函数逼近
On multiplicative Jacobi polynomials and function approximation through multiplicative series
AI总结：
本文引入乘法雅可比多项式，将经典雅可比多项式性质推广到乘法情形，并证明正实值函数可用乘法Jacobi-Fourier级数的部分乘积逼近。
摘要：
在本文中，我们引入了乘法雅可比多项式，它们是乘法Sturm-Liouville方程 \begin{equation*} \frac{d^*}{dx}\left( e^{(1-x^2)ω(x)}\odot \frac{d^*y}{dx} \right)\oplus \left(e^{ n(n+α+β+1)ω(x)}\odot y\right)=1, \\ x\in[-1,1], \end{equation*} 的解之一，其中 $ω(x)=(1-x)^α(1+x)^β$，$α, β>-1$ 为实数，$n$ 为非负整数。我们将经典雅可比多项式的一些性质推广到乘法情形。特别地，我们给出了乘法Legendre多项式以及第一类和第二类乘法Chebyshev多项式的若干性质。我们还证明了每个正实值函数都可以表示为乘法Jacobi-Fourier级数，并证明这类函数可以被这些级数的相应部分乘积所逼近。我们用一些例子说明了所得到的结果。
英文摘要：
In this contribution, we introduce the multiplicative Jacobi polynomials that arise as one of the solutions of the multiplicative Sturm-Liouville equation \begin{equation*} \frac{d^*}{dx}\left( e^{(1-x^2)ω(x)}\odot \frac{d^*y}{dx} \right)\oplus \left(e^{ n(n+α+β+1)ω(x)}\odot y\right)=1, \ x\in[-1,1], \end{equation*} where $ω(x)=(1-x)^α(1+x)^β$ with $α, β>-1$ real numbers and $n$ is a non-negative integer number. We extend some properties of classical Jacobi polynomials to the multiplicative case. In particular, we present several properties of multiplicative Legendre polynomials and multiplicative Chebyshev polynomials of first and second kind. We also prove that every real and positive function can be expressed as a multiplicative Jacobi-Fourier series and show that such functions can be approximated by the corresponding partial products of these series. We illustrate the obtained results with some examples.
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