We propose a new point of view in the study of Fourier analysis on graphs, taking advantage of localization in the Fourier domain. For a signal f on vertices of a weighted graph $\mathcal{G}$ with Laplacian matrix $\mathcal{L}$, standard Fourier analysis of f relies on the study of functions $g(\mathcal{L})f$ for some filters g on $I_{\mathcal{L}}$, the smallest interval containing the Laplacian spectrum $\mathrm{sp}(\mathcal{L})\subset I_{\mathcal{L}}$. We show that for carefully chosen partitions $I_{\mathcal{L}}={\bigsqcup }_{1\le k\le K}{I}_k$ ($I_k\subset I_{\mathcal{L}}$), there are many advantages in understanding the collection ${(g({\mathcal{L}}_{I_k})f)}_{1\le k\le K}$ instead of $g(\mathcal{L})f$ directly, where ${\mathcal{L}}_I$ is the projected matrix $P_I(\mathcal{L})\mathcal{L}$. First, the partition provides a convenient modelling for the study of theoretical properties of Fourier analysis and allows for new results in graph signal analysis (e.g. noise level estimation, Fourier support approximation). We extend the study of spectral graph wavelets to wavelets localized in the Fourier domain, called LocLets, and we show that well-known frames can be written in terms of LocLets. From a practical perspective, we highlight the interest of the proposed localized Fourier analysis through many experiments that show significant improvements in two different tasks on large graphs, noise level estimation and signal denoising. Moreover, efficient strategies permit to compute sequence ${(g({\mathcal{L}}_{I_k})f)}_{1\le k\le K}$ with the same time complexity as for the computation of $g(\mathcal{L})f$.

我们在图的傅里叶分析研究中提出了一个新的观点，利用傅里叶域中的定位。对于加权图顶点上的信号f$\mathcal{G}$ 拉普拉斯矩阵 $\mathcal{升}$, f 的标准傅立叶分析依赖于对函数的研究$G(\mathcal{升})F$对于一些过滤器克上${\mathrm{一世}}_{\mathcal{升}}$，包含拉普拉斯谱的最小区间 $\mathrm{sp}(\mathcal{升})\subset {\mathrm{一世}}_{\mathcal{升}}$. 我们证明对于精心选择的分区${\mathrm{一世}}_{\mathcal{升}}={\bigsqcup }_{1\le 克\le 钾}{\mathrm{一世}}_{克}$ (${\mathrm{一世}}_{克}\subset {\mathrm{一世}}_{\mathcal{升}}$)，理解集合有很多好处 ${(G({\mathcal{升}}_{{\mathrm{一世}}_{克}})F)}_{1\le 克\le 钾}$ 代替 $G(\mathcal{升})F$ 直接，哪里 ${\mathcal{升}}_{\mathrm{一世}}$ 是投影矩阵 ${磷}_{\mathrm{一世}}(\mathcal{升})\mathcal{升}$. 首先，分区为傅里叶分析的理论特性研究提供了方便的建模，并允许在图形信号分析中获得新结果（例如噪声水平估计、傅里叶支持近似）。我们将谱图小波的研究扩展到位于傅立叶域中的小波，称为 LocLets，并且我们证明了众所周知的帧可以用 LocLets 来写。从实践的角度来看，我们通过许多实验强调了所提出的局部傅里叶分析的兴趣，这些实验表明在大图的两个不同任务，噪声水平估计和信号去噪方面有显着改进。此外，有效的策略允许计算序列${(G({\mathcal{升}}_{{\mathrm{一世}}_{克}})F)}_{1\le 克\le 钾}$ 具有与计算相同的时间复杂度 $G(\mathcal{升})F$.