A Tauberian theorem and some of its applications

In general terms, a Tauberian theorem deals with the relationship between the properties of one transform of a measure with those of another transform. We will introduce the notion of a Tauberian theorm, and present our own recent theorem in this direction. Our theorem provides a uniform theory for the construction of certain localized kernels in a very general context. These in turn play a fundamental role in many different applications in numerical analysis, signal processing, and machine learning. We will discuss a few applications, for example, the construction of a theory inspired neural network for the solution of Burgers equation, inversion of Laplace transform of point masses, and an alternative theory for function approximation in the setting of diffusion geometry in machine learning without the need for any eigen-decomposition of a large matrix.