The speaker, Prof. Tsuyoshi Yoneda, introduced some studies about the approximation of functions by Deep Neural Networks (DNNs). The first topic was the approximation of an almost periodic function using a finite series constructed from a certain type of DNN. In the construction procedures, insights from both statistics and DNNs played a crucial role.

The second topic was the approximation of a simple function on a ball in higher-dimensional Euclidean space. In comparison with approximations using the Fourier series, his method based on DNNs has an advantage in pointwise convergence. In fact, Fourier series approximation does not perform well in higher-dimensional space, but his DNN-based method can provide an approximate approximation of the simple function, thereby achieving pointwise convergence.