Spectral Theory in Hilbert Space

15. SINGULAR PROBLEMS 109
Theorem 15.5.
(1) The integral
K F ∗ (y, t)W (y)u(y) dy converges in L2 P for u ∈
as K → I through compact subintervals of I. The limit
L2 W
is called the generalized Fourier transform of u and is
denoted by F(u) or û. We write this as û(t) = 〈u, F (·, t)〉W ,
although the integral may not converge pointwise.
(2) The mapping u ↦→ û has kernel H∞ and is unitary between HT
and L2 P so that the Parseval formula 〈u, v〉W = 〈û, ˆv〉P holds
if u, v ∈ L2 W and at least one of them is in HT .
(3) The integral
K F (x, t)dP (t)û(t) converges in HT as K → R
through compact intervals. If û = F(u) the limit is PT u, where
PT is the orthogonal projection onto HT . In particular the integral
is the inverse of the generalized Fourier transform on
HT . Again, we write u(x) = 〈û, F ∗ (x, ·)〉P for u ∈ HT , although
the integral may not converge pointwise.
(4) Let E∆ denote the spectral projector of ˜ T for the interval ∆.
Then E∆u(x) =
F (x, t) dP (t) û(t).
∆
(5) If (u, v) ∈ T then F(v)(t) = tû(t). Conversely, if û and tû(t)
are in L2 P , then F −1 (û) ∈ D(T ).
We will prove Theorem 15.5 through a sequence of lemmas. First
note that for u ∈ L2 W with compact support, the function û(λ) =
〈u, F (·, λ)〉W is an entire, matrix-valued function of λ since F (x, λ),
and thus also F ∗ (x, λ), is entire, locally uniformly in x, according to
Theorem 15.1.
Lemma 15.6. The function 〈Rλu, v〉W − ˆv ∗ (λ)M(λ)û(λ) is entire
for all u, v ∈ L2 W with compact supports.
Proof. If the supports are inside [a, b], direct calculation shows
that the function is
1
2
b
a
x a
−
x
b
v ∗ (x)W (x)F (x, λ)J −1 F ∗ (y, λ)W (y)u(y) dy dx .
This is obviously an entire function of λ.
As usual we denote the spectral projectors belonging to T (i.e.,
those belonging to ˜ T ) by Et.
Lemma 15.7. Let u ∈ L2 W have compact support and assume a < b
to be points of differentiability for both 〈Etu, u〉 and P (t). Then
(15.4) 〈Ebu, u〉 − 〈Eau, u〉 =
b
a
û ∗ (t) dP (t) û(t).