ABSTRACT
L'J,(T X S) = E I ® E2 $ El ® F~ $ F1 ® E2 E9 Fl®F~. Obviously, the spaces E1®E2 , E1®F2, and Fl®E2 are invariant under the operator (14.1) with the kernel (15.45), and Fl~F'J, is mapped by this operator to the trivial functions. Consequently, equation (15.32) may be replaced by the equations
(15.47)
(15.48)
h ffi . (A:) (A:) (k ) d b(A:) &.CA:)Moreover, t e coe Clents a1 , ••• , ap = 1,2 an 1"'" u/I (k = 1,3) are given by (15.50) a~k)(a) = lr z(k)(T, a}tPi(T}dJ£(T) (i = 1, ... ,p) and
(15.51) h~A:)(t) = Is z(A:)(t, u)J£j(u) dv(u) (j = 1, ... , q). Let us consider first the equation (15.46). Putting (15.46) into (15.50) for k =1 and into (15.51) for k = 1 we obtain the equations
1=1 j=l for i = 1, ... ,p, and
for j = 1, ... , q, where the constants Cil and djA: and the functions ll) and gj1) are defined by
(15.55)
and
and the unknown constants zH) and yJ:) may be determined by
{
z(9 = I b(I)(T),p,(T)d/J(T), 'J iT J
(15.56) y(:) = I afl)( (1 )/Jj(U)dll((1).
