The purpose of this paper is to clarify some mathematical issues associated
with the direct-interaction approximation (DIA) given by Kraichnan [Phys.
Rev. 109 (1958), 1407; J. Fluid Mech. 5 (1959), 497; Phys. Fluids 7 (1964),
1723; 8 (1965), 575; 9 (1966), 1728] via its application to Kraichnan's [J
. Math. Phys. 2 (1961), 124] model equation. The latter process is shown to
lead to a functional equation; the basic properties and construction of th
e solution of this equation are addressed. The relation of the DLA solution
to the perturbative solution is discussed within the framework of a contin
ued-fraction representation. (C) 1999 Academic Press.