Lemma 58.21.1Suppose M is an n × n matrix. Suppose also that
α∗M α = 0
for allα∈ ℂ^{n}. Then M = 0.
Proof:Suppose λ is an eigenvalue for M and let α be an associated eigenvector.
0 = α ∗M α = α∗λα = λα∗α = λ |α |2
and so all the eigenvalues of M equal zero. By Schur’s theorem there is a unitary matrix
U such that
( 0 ∗ )
| . 1| ∗
M = U ( .. ) U (58.21.51)
0 0
(58.21.51)
where the matrix in the middle has zeros down the main diagonal and zeros below the
main diagonal. Thus
( )
0 0
M ∗ = U|( ... |) U∗
∗ 0
2
where M^{∗} has zeros down the main diagonal and zeros above the main diagonal. Also
taking the adjoint of the given equation for M, it follows that for all α,
∗ ∗
α M α = 0
Therefore, M + M^{∗} is Hermitian and has the property that
α∗(M + M ∗)α = 0.
Thus M + M^{∗} = 0 because it is unitarily similar to a diagonal matrix and the above
equation can only hold for all α if M + M^{∗} has all zero eigenvalues which implies the
diagonal matrix has zeros down the main diagonal. Therefore, from the formulas for
M,M^{∗},
and so the sum of the two matrices in the middle must also equal 0. Hence the entries
of the matrix in the middle in 58.21.51 are all equal to zero. Thus M = 0 as
claimed.
Definition 58.21.2A Borel measurable function, f : ℝ^{n}→ ℂ is called positive definiteif whenever
{tk}
_{k=1}^{p}⊆ℝ^{n},α∈ℂ^{p}
∑
f (tj − tk)αjαk-≥ 0 (58.21.52)
k,j
(58.21.52)
The first thing to notice about a positive definite function is the following which
implies these functions are automatically bounded.
Lemma 58.21.3If f is positive definite then whenever
Therefore, the Cauchy Schwarz inequality holds for
[⋅,⋅]
and it follows
1∕2 1∕2
|[α, β]| = |(Fα, β)| ≤ (F α,α ) (Fβ, β) .
Letting α = e_{k} and β = e_{j}, it follows F_{ss}≥ 0 for all s and
|Fkj| ≤ F1∕2F 1∕2
kk jj
which says nothing more than
|f (t − t)| ≤ f (0)1∕2f (0)1∕2 = f (0).
j k
This proves the lemma.
With this information, here is another useful lemma involving positive definite
functions. It is interesting because it looks like the formula which defines what it means
for the function to be positive definite.
Lemma 58.21.4Let f be a positive definite function as defined above and let μ be afinite Borel measure. Then
∫ ∫
f (x− y)dμ (x)dμ(y) ≥ 0. (58.21.54)
ℝn ℝn
(58.21.54)
If μ also has the property that it is symmetric, μ
(F )
= μ
(− F)
for all F Borel,then
∫
n f (x)d (μ ∗μ)(x) ≥ 0. (58.21.55)
ℝ
(58.21.55)
Proof: By definition if
{tj}
_{j=1}^{p}⊆ ℝ^{n}, and letting α =
(1,⋅⋅⋅,1)
^{T}∈ ℝ^{n},
∑
f (tj − tk) ≥ 0.
j,k
Therefore, integrating over each of the variables,
because it is just a complex number times its conjugate.
Thus every characteristic function is continuous, equals 1 at 0, and is positive definite.
Bochner’s theorem goes the other direction.
To begin with, suppose μ is a finite measure on ℬ
(ℝn)
. Then for S the Schwartz
class, μ can be considered to be in the space of linear transformations defined on S, S^{∗}
as follows.
∫
μ(f) ≡ fdμ.
Recall F^{−1}
(μ)
is defined as
∫
F− 1(μ)(f ) ≡ μ(F −1f) = F−1fdμ
ℝn
∫ ∫
---1--- ix⋅y
= (2π)n∕2 ℝn ℝn e f (y)dydμ
∫ ( ∫ )
= ---1--- eix⋅ydμ f (y) dy
ℝn (2π)n∕2 ℝn
and so F^{−1}
(μ)
is the bounded continuous function
1 ∫ ix⋅y
y → ---n∕2- n e dμ.
(2π) ℝ
Now the following lemma has the main ideas for Bochner’s theorem.
Lemma 58.21.6Suppose ψ
(t)
is positive definite,t →ψ
(t)
is in L^{1}
(ℝn, mn)
wherem_{n}is Lebesgue measure, ψ
(0)
= 1, and ψ is continuous at 0. Then there exists a uniqueprobability measure, μ defined on the Borel sets of ℝ^{n}such that
ϕμ (t) = ψ(t).
Proof: If the conclusion is true, then
∫
ψ (t) = eit⋅xdμ(x) = (2π)n∕2F−1 (μ )(t).
ℝn
Recall that μ ∈S^{∗}, the algebraic dual of S . Therefore, in S^{∗},
--1---F (ψ) = μ.
(2π )n∕2
That is, for all f ∈S,
∫ ∫
f (y)dμ(y) = ---1--- F (ψ)(y)f (y)dy
ℝn (2π)n∕2 ℝn
--1-- ∫ ( ∫ −iy⋅x )
= (2π)n ℝn f (y) ℝn e ψ (x)dx dy.(58.21.56)
I will show
∫ (∫ )
f → --1n- f (y ) e−iy⋅xψ (x)dx dy
(2π ) ℝn ℝn
is a positive linear functional and then it will follow from 58.21.56 that μ is unique. Thus
it is needed to show the inside integral in 58.21.56 is nonnegative. First note that the
integrand is a positive definite function of x for each fixed y. This follows from