Page 557 - Handbook Of Integral Equations

and
|λ 1 |≤|λ 2 |≤ ··· ≤|λ n |≤ ··· . (5)

If there are infinitely many characteristic values, then it follows from the fourth Fredholm
theorem that their only accumulation point is the point at infinity, and hence λ n →∞ as n →∞.
The set of all characteristic values and the corresponding normalized eigenfunctions of a sym-
metric kernel is called the system of characteristic values and eigenfunctions of the kernel. The
system of eigenfunctions is said to be incomplete if there exists a nonzero square integrable function
that is orthogonal to all functions of the system. Otherwise, the system of eigenfunctions is said to
be complete.

11.6-2. Bilinear Series
Assume that a kernel K(x, t) admits an expansion in a uniformly convergent series with respect to
the orthonormal system of its eigenfunctions:

∞

K(x, t)= a k (x)ϕ k (t) (6)
k=1
for all x in the case of a continuous kernel or for almost all x in the case of a square integrable
kernel.
We have
b ϕ k (x)

a k (x)= K(x, t)ϕ k (t) dt = , (7)
a λ k
and hence
∞
ϕ k (x)ϕ k (t)
K(x, t)= . (8)
λ k
k=1
Conversely, if the series
∞
ϕ k (x)ϕ k (t)

(9)
λ k
k=1
is uniformly convergent, then
∞
ϕ k (x)ϕ k (t)
K(x, t)= .
λ k
k=1
The following assertion holds: the bilinear series (9) converges in mean-square to the ker-
nel K(x, t).
If a symmetric kernel K(x, t) has finitely many characteristic values, then it is degenerate,
because in this case we have
n
ϕ k (x)ϕ k (t)
K(x, t)= . (10)
λ k
k=1
A kernel K(x, t) is said to be positive definite if for all functions ϕ(x) that are not identically
zero we have
b b
K(x, t)ϕ(x)ϕ(t) dx dt >0,
a a
and the above quadratic functional vanishes for ϕ(x)=0 only. Such a kernel has positive characteristic
values only. A negative definite kernel is defined similarly.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 540