Measure, Integration & Real Analysis, 2021a

370 Chapter 11 Fourier Analysis
Our next result shows that multiplication of Fourier transforms corresponds to
convolution of the corresponding functions.
11.66 Fourier transform of a convolution
Suppose f , g ∈ L 1 (R). Then
for every t ∈ R.
( f ∗ g)̂(t) = ̂f (t) ĝ(t)
Proof Adjust the proof of 11.44 to the context of R.
Poisson Kernel on Upper Half-Plane
As usual, we identify R 2 with C, as illustrated in the following definition. We will
see that the upper half-plane plays a role in the context of R similar to the role that
the open unit disk plays in the context of ∂D.
11.67 Definition H; upper half-plane
• H denotes the open upper half-plane in R 2 :
H = {(x, y) ∈ R 2 : y > 0} = {z ∈ C :Imz > 0}.
• ∂H is identified with the real line:
∂H = {(x, y) ∈ R 2 : y = 0} = {z ∈ C :Imz = 0} = R.
Recall that we defined a family of functions on ∂D called the Poisson kernel on D
(see 11.14, where the family is called the Poisson kernel on D because 0 ≤ r < 1 and
ζ ∈ ∂D implies rζ ∈ D). Now we are ready to define a family of functions on R that
is called the Poisson kernel on H [because x ∈ R and y > 0 implies (x, y) ∈ H].
The following definition is motivated by 11.62. The notation P r for the Poisson
kernel on D and P y for the Poisson kernel on H is potentially ambiguous (what is
P 1/2 ?), but the intended meaning should always be clear from the context.
11.68 Definition P y ; Poisson kernel
• For y > 0, define P y : R → (0, ∞) by
P y (x) = 1 π
y
x 2 + y 2 .
• The family of functions {P y } y>0 is called the Poisson kernel on H.
Measure, Integration & Real Analysis, by Sheldon Axler