论文标题
关于$ l _ {[ - 1,1]}^{p，（η，τ）} $ space in $ l _ {[ - 1,1]} $ jacobi系列的收敛性
On the Convergence of Random Fourier--Jacobi Series in $L_{[-1,1]}^{p,(η,τ)}$ space
论文作者
论文摘要
刘和刘引入了随机傅立叶变换，这是Hermite函数中的一个随机傅立叶序列，并将其应用于图像加密和解密。他们期望其在光学和信息技术中的应用。这些促使我们研究正交多项式中的随机傅立叶系列。最近，我们已经建立了随机傅立叶 - 雅各比系列$ \ sum \ limits_ {n = 0}^\ infty d_n r_n r_n（ω）φ_n（y），$φ_n（y）$是正常的jacobi jacobi jacobi jacobi jacobi polynomials $ p_n^$ p_n^$ eysup $ eytem $ eytemudemem $ eytemudememe与随机过程相关联的变量，例如Wiener过程，对称稳定过程和标量$ d_n $是某些连续函数类别中函数的傅立叶 - jacobi系数。据观察，随机系列的收敛方式取决于标量$ d_n $和随机过程的选择。在本文中，我们调查了标量$ d_n，$被选为傅立叶 - 雅各比（Jacobi）的功能系数，在某些加权的$ l _ {[ - 1,1]}^{p，（η，τ）} $ space中，以便随机系列收敛。此外，还研究了总和函数的连续性属性。
Liu and Liu introduced the random Fourier transform, which is a random Fourier series in Hermite functions, and applied it to image encryption and decryption. They expected its applications in optics and information technology. These motivated us to look into random Fourier series in orthogonal polynomials. Recently, we have established the convergence of random Fourier--Jacobi series $\sum\limits_{n=0}^\infty d_n r_n(ω)φ_n(y),$ where $φ_n(y)$ are the orthonormal Jacobi polynomials $p_n^{(γ,δ)}(y),$ $r_n(ω)$ are random variables associated with stochastic processes like the Wiener process, the symmetric stable process, and the scalars $d_n$ are the Fourier--Jacobi coefficients of functions in some classes of continuous functions. It is observed that the mode of convergence of the random series depends on the choice of the scalars $d_n$ and the stochastic processes. In this article, we have investigated the scalars $d_n,$ which are chosen to be the Fourier--Jacobi coefficients of functions in some weighted $L_{[-1,1]}^{p,(η,τ)}$ spaces, so that the random series converges. Further, the continuity property of the sum functions is studied.
