The object of this paper is
to characterize functions which have L^{2} expansions in terms of polynomial solutions
P_{n,ν}(x,t) of the generalized heat equation
 (*) 
and in terms of the Appell transforms W_{n,ν}(x,t) of the P_{n,ν}(x,t). H^{∗} denotes
the C^{2} class of functions u(x,t) which, for a < t < b, satisfy (*) and for
which
for all t, t′, a < t′ < t < b, 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) ∈ H^{∗}, − σ ≦ t < 0, and
for each fixed t − σ ≦ t < 0, 0 ≦ x < ∞. Then, for −σ ≦ t < 0,
and
where
and
Theorem. If u(x,t) ∈ H^{∗}, 0 < t ≦ σ, and if
for each fixed t, 0 < t ≦ σ, 0 ≦ x < ∞, then, for 0 < t ≦ σ,
and
where b_{n} is given above and
Theorem. If u(x,t) ∈ H^{∗}, 0 < σ ≦ t, and if
for each fixed t, 0 < σ ≦ t, 0 ≦ x < ∞, then, for 0 < σ ≦ t,
and
where b_{n} is given above and
