On the Completeness of Gaussians in a Hilbert Functional Space

摘要

Let w(T) (t) and w(Omega) (omega) be non-negative functions defined for t is an element of R and omega is an element of R, where R = (-infinity, infinity). Some regularity conditions are posed on these functions. The space H-wT,H-w Omega consists of those functions x for which integral(infinity)(-infinity) vertical bar x(t)vertical bar(2)w(T) (t)dt + integral(infinity)(-infinity) vertical bar(x) over cap(omega)vertical bar(2)w Omega (omega)d omega = parallel to x parallel to(2)(H wT,w Omega) < infinity, where <(x)over cap> is the Fourier transform of the function x. We show that the system of Gaussians {exp(-alpha(t - tau)(2))}, where alpha runs over R+ = (0, +infinity) and tau runs over R, is a complete system in the space H-wT,H-w Omega.