Analysis of functions on combinatorial graphs is an emerging field attracting more and more attention. In this paper, we study the approximation of functions defined on combinatorial graphs by functions in Paley–Wiener spaces. First, we use a family of graph translation operators to define the modulus of smoothness, which has several properties similar to their counterparts in the classical approximation theory. Next, we establish Jackson’s and Bernstein’s inequalities for functions defined on graphs. Finally, we provide an estimation on the decay of graph Fourier coefficients in terms of the modulus of smoothness. These results lead to a theory of approximation of functions on combinatorial graphs and have potential applications to filtering, denoising, data dimension reduction, image processing and learning theory.

组合图上的函数分析是一个越来越受到关注的新兴领域。在本文中，我们研究了由 Paley-Wiener 空间中的函数定义在组合图上的函数的逼近。首先，我们使用一系列图平移算子来定义平滑度模数，它具有几个与经典近似理论中的对应物相似的属性。接下来，我们为在图上定义的函数建立 Jackson 和 Bernstein 不等式。最后，我们根据平滑度模数提供了对图傅立叶系数衰减的估计。这些结果导致了组合图上函数逼近的理论，并且在过滤、去噪、数据降维、图像处理和学习理论方面具有潜在的应用价值。