Page 558 - Handbook Of Integral Equations

Each symmetric positive definite (or negative definite) continuous kernel can be decomposed in
a bilinear series in eigenfunctions that is absolutely and uniformly convergent with respect to the
variables x, t.
The assertion remains valid if we assume that the kernel has finitely many negative (positive,
respectively) characteristic values.
If a kernel K(x, t) is symmetric, continuous on the square S = {a ≤ x ≤ b, a ≤ t ≤ b}, and has
uniformly bounded partial derivatives on this square, then this kernel can be expanded in a uniformly
convergent bilinear series in eigenfunctions.

11.6-3. The Hilbert–Schmidt Theorem
If a function f(x) can be represented in the form
b
f(x)= K(x, t)g(t) dt, (11)
a
where the symmetric kernel K(x, t) is square integrable and g(t) is a square integrable function,
then f(x) can be represented by its Fourier series with respect to the orthonormal system of
eigenfunctions of the kernel K(x, t):

∞

f(x)= a k ϕ k (x), (12)
k=1
where
b

a k = f(x)ϕ k (x) dx, k =1, 2, ...
a
Moreover, if
b

2
K (x, t) dt ≤ A < ∞, (13)
a
then the series (12) is absolutely and uniformly convergent for any function f(x) of the form (11).
Remark 1. In the Hilbert–Schmidt theorem, the completeness of the system of eigenfunctions
is not assumed.

11.6-4. Bilinear Series of Iterated Kernels
By the definition of the iterated kernels, we have
b
K m (x, t)= K(x, z)K m–1 (z, t) dz, m =2, 3, ... (14)
a
The Fourier coefficients a k (t) of the kernel K m (x, t), regarded as a function of the variable x, with
respect to the orthonormal system of eigenfunctions of the kernel K(x, t) are equal to
b ϕ k (t)

a k (t)= K m (x, t)ϕ k (x) dx = m . (15)
a λ k
On applying the Hilbert–Schmidt theorem to (14), we obtain
∞
ϕ k (x)ϕ k (t)
K m (x, t)= m , m =2, 3, ... (16)
λ
k=1 k
In formula (16), the sum of the series is understood as the limit in mean-square. If in addition to the
above assumptions, inequality (13) is satisfied, then the series in (16) is uniformly convergent.

© 1998 by CRC Press LLC

© 1998 by CRC Press LLC
Page 541