论文标题
圆环和应用上的分数傅立叶系列的收敛
Convergence of fractional Fourier series on the torus and applications
论文作者
论文摘要
在本文中，我们介绍了分数圆环的分数傅立叶序列，并研究了分数傅立叶序列的一些基本事实，例如分数卷积和分数近似。同时，还给出了分数傅立叶反转和泊松求和公式。我们进一步讨论了分数傅立叶系数的衰减与函数的平滑度之间的关系。使用分数FEJER内核的性能，可以建立分数傅立叶系列的侧面收敛。最后，我们介绍了分数傅立叶序列到具有周期性边界条件的分数偏微分方程的应用。此外，我们在分数圆环上应用近似方法来恢复非平稳信号。
In this paper, we introduce the fractional Fourier series on the fractional torus and study some basic facts of fractional Fourier series, such as fractional convolution and fractional approximation. Meanwhile, fractional Fourier inversion and Poisson summation formula are also given. We further discuss the relationship between the decay of fractional Fourier coefficients and the smoothness of a function. Using the properties of fractional Fejer kernel, the pointwise convergence of fractional Fourier series can be established. Finally, we present the applications of fractional Fourier series to fractional partial differential equations with periodic boundary condition. Moreover, we apply approximation methods on the fractional torus to recover the non-stationary signals.
