发表机构
Sun Yat-Sen University; Università degli Studi di Napoli Parthenope（中山大学; 那不勒斯帕尔泰诺佩大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明满足变差型条件的二元函数的切比雪夫展开均匀逼近定理，并应用于图上时变信号的谱图小波变换，实现稳定重构并在去噪中优于固定参数方法。
AI 中文摘要
我们证明了关于两个变量的连续函数的切比雪夫展开的均匀逼近定理。在正方形$[-1,1]^2$上的变差型条件下，一个连续函数在第一变量上允许一个均匀收敛的切比雪夫展开，其系数是第二变量的连续函数。这些结果被应用于有限加权图上时变信号的谱图小波变换：缩放核和小波核通过具有时变系数的展开来逼近，这些展开的次数不依赖于时间；近似变换与其伴随的复合在时间无关情形下具有相同的显式系数公式；通过伪逆的重构是稳定的，并具有以均匀核误差表示的显式界。在传感器网络上的数值实验证实了收敛性和稳定性估计。在去噪问题中，使用时变变换参数对图小波系数进行软阈值处理优于其时间无关的固定参数对应方法。
英文摘要
We prove uniform approximation theorems for Chebyshev expansions of continuous functions of two variables. Under a variation-type condition on the square $[-1,1]^2$, a continuous function admits a uniformly convergent Chebyshev expansion in the first variable whose coefficients are continuous functions of the second variable. These results are applied to the spectral graph wavelet transform of time-varying signals on finite weighted graphs: the scaling and wavelet kernels are approximated by expansions with time-varying coefficients whose degrees do not depend on time, the composition of the approximate transform with its adjoint admits the same explicit coefficient formulas as in the time-independent case, and reconstruction by the pseudoinverse is stable, with explicit bounds in terms of the uniform kernel errors. Numerical experiments on a sensor network confirm the convergence and stability estimates. In a denoising problem, soft thresholding of the graph wavelet coefficients with a time-varying transform parameter improves over its time-independent, fixed-parameter counterpart.
Comments32 pages, 2 figures, 2 tables
