59.21. POSITIVE DEFINITE FUNCTIONS, BOCHNER’S THEOREM 1941

The integrand is of the formβ∗

 1...1

β

∗

 1...1

≥ 0

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∗ asfollows.

µ ( f )≡∫

f dµ.

Recall F−1 (µ) is defined as

F−1 (µ)( f )≡ µ(F−1 f

)=∫Rn

F−1 f dµ

=1

(2π)n/2

∫Rn

∫Rn

eix·y f (y)dydµ

=∫Rn

(1

(2π)n/2

∫Rn

eix·ydµ

)f (y)dy

and so F−1 (µ) is the bounded continuous function

y→ 1

(2π)n/2

∫Rn

eix·ydµ.

Now the following lemma has the main ideas for Bochner’s theorem.

Lemma 59.21.6 Suppose ψ (t) is positive definite, t→ ψ (t) is in L1 (Rn,mn) where mnis Lebesgue measure, ψ (0) = 1, and ψ is continuous at 0. Then there exists a uniqueprobability measure, µ defined on the Borel sets of Rn such that

φ µ (t) = ψ (t) .

Proof: If the conclusion is true, then

ψ (t) =∫Rn

eit·xdµ (x) = (2π)n/2 F−1 (µ)(t) .

Recall that µ ∈S∗, the algebraic dual of S . Therefore, in S∗,

1

(2π)n/2 F (ψ) = µ.