58.21 Positive Definite Functions, Bochner’s Theorem
First here is a nice little lemma about matrices.
Lemma 58.21.1 Suppose M is an n × n matrix. Suppose also that
for all α ∈ ℂn. Then M = 0.
Proof: Suppose λ is an eigenvalue for M and let α be an associated eigenvector.
and so all the eigenvalues of M equal zero. By Schur’s theorem there is a unitary matrix
U such that
where the matrix in the middle has zeros down the main diagonal and zeros below the
main diagonal. Thus
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,
Therefore, M + M∗ is Hermitian and has the property that
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
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
Definition 58.21.2 A Borel measurable function, f : ℝn → ℂ is called positive definite
The first thing to notice about a positive definite function is the following which
implies these functions are automatically bounded.
Lemma 58.21.3 If f is positive definite then whenever
k=1p are p points in
. In particular, for all t,
Proof: Let F be the p × p matrix such that
Then 58.21.52 is of the form
where this is the inner product in ℂp. Letting
it is obvious
I claim it also satisfies
To verify this last claim, note that since α∗Fα is real,
and so for all α ∈ ℂp,
which from Lemma 58.21.1 implies F∗ = F. Hence F is self adjoint and it follows
Therefore, the Cauchy Schwarz inequality holds for
and it follows
Letting α = ek and β = ej, it follows Fss ≥ 0 for all s and
which says nothing more than
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 58.21.4 Let f be a positive definite function as defined above and let μ be a
finite Borel measure. Then
If μ also has the property that it is symmetric, μ
for all F Borel,
Proof: By definition if
j=1p ⊆ ℝn,
and letting α
T ∈ ℝn,
Therefore, integrating over each of the variables,
Dividing both sides by p
which shows 58.21.54.
To verify 58.21.55, use 58.14.25.
and since μ is symmetric, this equals
by the first part of the lemma. This proves the lemma.
Lemma 58.21.5 Let μt be the measure defined on ℬ
for t > 0. Then μt ∗ μt = μ2t and each μt is a probability measure.
Proof: By Theorem 58.14.7,
Each μt is a probability measure because it is the distribution of a normally distributed
random variable of mean 0 and covariance tI.
Now let μ be a probability measure on ℬ
and so by the dominated convergence theorem, ϕμ is continuous and also ϕμ
is also positive definite. Let α ∈ ℂp
a sequence of points of ℝn.
Now let β
Then the above equals
The integrand is of the form
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
Then for S
can be considered to be in the space of linear transformations defined on S
is defined as
and so F−1
is the bounded continuous function
Now the following lemma has the main ideas for Bochner’s theorem.
Lemma 58.21.6 Suppose ψ
is positive definite, t →ψ
is in L1
mn is Lebesgue measure, ψ
, and ψ is continuous at 0. Then there exists a unique
probability measure, μ defined on the Borel sets of ℝn such that
Proof: If the conclusion is true, then
Recall that μ ∈S∗, the algebraic dual of S . Therefore, in S∗,
That is, for all f ∈S,
I will show
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
Let t > 0 and
Then by dominated convergence theorem,
Letting dη2t = h2t
it follows from Lemma 58.21.5 η2t
= ηt ∗ηt
and since these are
symmetric measures, it follows from Lemma 58.21.4
the above equals
Thus the above functional is a positive linear functional and so there exists a unique
Radon measure, μ satisfying
for all f ∈ Cc
. Thus from the dominated convergence theorem, the above holds for
also. Hence for all f ∈S
and considering μ
as an element of S∗,
It follows that in S∗,
in L1. Since the right side is continuous and the left is given continuous at t = 0 and
equal to 1 there, it follows
and so μ is a probability measure as claimed. This proves the lemma.
The following is Bochner’s theorem.
Theorem 58.21.7 Let ψ be positive definite, continuous at 0, and ψ
there exists a unique Radon probability measure μ such that ψ
Proof: If ψ ∈ L1
then the result follows from Lemma 58.21.6
. By Lemma
is bounded. Consider
, x →ψt
is continuous at
and ψt ∈ L1