On the Euclidean domains of classical signal processing, linking of signal samples to underlying coordinate structures is straightforward. While graph adjacency matrices totally define the quantitative associations among the underlying graph vertices, a major problem in graph signal processing is the lack of explicit association of vertices with an underlying coordinate structure. To make this link, we propose an operation, called the vertex multiplication (VM) , which is defined for graphs and can operate on graph signals. VM, which generalizes the coordinate multiplication (CM) operation in time series signals, can be interpreted as an operator that assigns a coordinate structure to a graph. By using the graph domain extension of differentiation and graph Fourier transform (GFT), VM is defined such that it shows Fourier duality that differentiation and CM operations are duals of each other under Fourier transformation (FT). Numerical examples and applications are also presented.

中文翻译：

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图信号处理：顶点乘法

在经典信号处理的欧几里得域中，将信号样本链接到基础坐标结构是直接的。虽然图邻接矩阵完全定义了底层图顶点之间的定量关联，但图信号处理中的一个主要问题是缺乏顶点与底层坐标结构的显式关联。为了建立这种联系，我们提出了一项名为顶点乘法 (VM) ，它是为图形定义的，可以对图形信号进行操作。VM 概括了时间序列信号中的坐标乘法 (CM) 操作，可以将其解释为将坐标结构分配给图形的运算符。通过使用微分和图傅里叶变换（GFT）的图域扩展，VM被定义为显示傅里叶对偶性，即微分和CM运算在傅里叶变换（FT）下互为对偶。还提供了数值示例和应用。