LOGARITHMIC INTEGRALS, SOBOLEV SPACES AND RADON TRANSFORM IN THE PLANE

Dang Vu Giang

Abstract

           We prove that the set {φ₀, φ₁, φ₄, ... φ_3k+1, ...} of Hermite functions is an orthogonal system in the Sobolev space H¹(R) = H₍₁₎(R). Furthermore, the logarithmic integral of a function f from the real Hardy space H¹(R) is exactly the primitive function of - (the Hilbert transform of f ). And more interesting formulas are found for Radon transform of Hermite-like functions.