论文标题
关于随机傅里叶 - 雅各比的收敛性系列连续函数
On the Convergence of Random Fourier-Jacobi Series of Continuous functions
论文作者
论文摘要
在众多科学分支中，对正交多项式和随机傅立叶系列的兴趣以及对正交多项式中随机傅立叶序列的一些研究激发了我们专注于Jacobi多项式中的随机傅立叶系列。在本说明中，已经尝试研究某些随机雅各比系列的随机收敛性。我们研究了随机系列$ \ sum_ {n = 0}^\ infty d_n r_n（ω）φ_n（y）$在正交多项式中$φ_n（y）$带随机变量$ r_n（ω）。稳定的过程和维纳过程。 $φ_n（y）$被选择为jacobi多项式及其变体，具体取决于与随机过程相关的随机变量。随机串联的收敛是针对雅各比多项式的不同参数的$γ$，具有标量$ d_n $的相应选择，这是合适的连续函数类别的傅里叶 - 雅各比系数。观察到与连续随机过程相关的随机傅立叶 - 雅各比序列的总和函数是随机积分。还讨论了总和函数的连续性属性。
The interest in orthogonal polynomials and random Fourier series in numerous branches of science and a few studies on random Fourier series in orthogonal polynomials inspired us to focus on random Fourier series in Jacobi polynomials. In the present note, an attempt has been made to investigate the stochastic convergence of some random Jacobi series. We looked into the random series $\sum_{n=0}^\infty d_n r_n(ω)φ_n(y)$ in orthogonal polynomials $φ_n(y)$ with random variables $r_n(ω).$ The random coefficients $r_n(ω)$ are the Fourier-Jacobi coefficients of continuous stochastic processes such as symmetric stable process and Wiener process. The $φ_n(y)$ are chosen to be the Jacobi polynomials and their variants depending on the random variables associated with the kind of stochastic process. The convergence of random series is established for different parameters $γ,δ$ of the Jacobi polynomials with corresponding choice of the scalars $d_n$ which are Fourier-Jacobi coefficients of a suitable class of continuous functions. The sum functions of the random Fourier-Jacobi series associated with continuous stochastic processes are observed to be the stochastic integrals. The continuity properties of the sum functions are also discussed.
