Convergence in mean of some random Fourier series

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• 作者：Saroj Kumar Dash ; Swadheenan ; a Pattanayak
• 关键词：Symmetric stable process ; Random Fourier– ; Stieltjes series ; Stochastic integral ; Fractional integral
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2008
• 出版时间：1 March 2008
• 年：2008
• 卷：339
• 期：1
• 页码：98-107
• 全文大小：134 K

文摘

For a symmetric stable process X(t,ω) with index α(1,2], fL^p[0,2π], pα, and , we establish that the random Fourier–Stieltjes (RFS) series converges in the mean to the stochastic integral , where f_β is the fractional integral of order β of the function f for . Further it is proved that the RFS series is Abel summable to . Also we define fractional derivative of the sum of order β for a_n, A_n(ω) as above and . We have shown that the formal fractional derivative of the series of order β exists in the sense of mean.