Hilbert 13th's Problem and Quasi-Interpolation

In 1900, David Hilbert raised a conjecture known as Hilbert's 13th problem. In 1957, Kolmogorov and his 19 year student Arnold gave a negative answer to Hilbert's 13th problem by proving a remarkable fact that any multivariate function can be represented as addition and composition of (finite) numbers of univariate functions, which was called Kolmogorov’s superposition theorem. There have been many attempts to generalized Kolmogorov’s superposition theorem, such as radial basis function approximation, Kronecker-product approximation, additive models, neural networks as well as deep learning. However, most attempts require solving a minimization problem to obtain the final approximant. In this talk, I shall provide our works on circumventing the problem under the framework of quasi-interpolation. In addition, I shall provide some of our works on constructions of quasi-interpolation, properties and applications of quasi-interpolation.