Linear partial differential equations and real analytic approximations of rough functions

Abstract

Many common approximation methods exist such as linear or polynomial interpolation, splines, Taylor series, or generalized Fourier series. Unfortunately, many of these approximations are not analytic functions on the entire real line, and those that are diverge at infinity and therefore are only valid on a closed interval or for compactly supported functions. Our method takes advantage of the smoothing properties of certain linear partial differential equations to obtain an approximation which is real analytic, converges to the function on the entire real line, and yields particular conservation laws. This approximation method applies to any L₂ function on the real line which may have some rough behavior such as discontinuities or points of nondifferentiability. For comparison, we consider the well-known Fourier-Hermite series approximation. Finally, for some example functions the approximations are found and plotted numerically.