Local approximation using Hermite functions

Understand

We develop a wavelet like representation of functions in $L^p(\mathbb{R})$ based on their Fourier--Hermite coefficients; i.e., we describe an expansion of such functions where the local behavior of the terms characterize completely the local smoothness of the target function.

• In the case of continuous functions, a similar expansion is given based on the values of the functions at arbitrary points on the real line.
• In the process, we give new proofs for the localization of certain kernels, as well as some very classical estimates such as the Markov--Bernstein inequality.