<i>L</i>²expansions in terms of generalized heat polynomials and of their Appell transforms

Abstract

The object of this paper is to characterize functions which have L 2 expansions in terms of polynomial solutions Pn,*(x, t) of the generalized heat equation ( * J!_ + 2L J- u (r t) -JL u(x f) L dx 2 x dx J at and in terms of the Appell transforms W n ,(x, t) of the P n ,(x, t). H* denotes the C 2 class of functions u(x, t) which, for a<t<b, satisfy (*) and for which G(x, y\ tt')u(y, t f )d(y), 0 d(x) = 2W*^\r(v + for all t, V, a < V < t < 6, the integral converging absolutely, where G(x, y; t) is the source solution of (*). The principal results are the following: THEOREM. Let u(x, t)eH*, -^t<0, and u(x, t)[G(x; -t) e L 2 for each fixed t -^t<0, 0 ^ x < oo. Then, for - ^ t < 0, Jo where 6 W = p^w!]" 1and oo n,v(y, -t)d{y) .