§ 3. Some Applications of Cauchy’s Method
103
Theorem 13. Let k be a real number. The space H
1
k (P ) of holomorphic
functions such that
Im(z)>0
|f (z)| y
k−2 dxdy < +∞
(12.24)
does not reduce to {0} if and only if k > 1. Any holomorphic solution of (24)
is the complex Fourier transform of a continuous function ˆ
f (t) which is zero
for t < 0. Formulas (17), (18) and (19) hold.
Exercise 6. Calculate the function ˆ
f (t) corresponding to (23). p need
not be assumed to be an integer nor a real since function (24) has uniform
branches on y > 0.
To conclude this section, note that, while we have obtained important
properties of the functions ˆ
f , we have not characterized them as we did in
the Paley-Wiener theorem; as far as I know – given the flood of publications
since the last fifty years, it is necessary to be prudent –, the answer to this
question will never be known. It is, however, fully known if the functions f (z)
are taken to be square integrable on the half-plane: these are complex Fourier
transforms of the functions ˆ
f (t) which are zero for t ≤ 0 and for which
+∞
0
ˆ
f (t)
2
t
1−k dt < +∞ ;
but these are now functions on the Lebesgue L
2 space and the result cannot
be obtained without the help of the whole theory of Fourier transforms on
L
2 . This topic will be presented in Chap. XI.
13 – The Mellin Transform
(i) Questions of convergence. To obtain Paley-Wiener type theorems that hold
for meromorphic functions rather than holomorphic ones as in the previous
n
◦ – an important problem in analytic number theory for example –, it is
useful to reformulate the complex Fourier transform. If the change of variable
exp(2πt) = u is carried out in integral (12.1) which it is defined by, then u > 0
and
2πdt = du/u = d
∗ u .
Setting iz = s, we get e(tz) = u
s and ˆ
f (t) = F (e
2πt ), and so
2πf (z) =
+∞
0
F (u)u
s d
∗ u .
We often come across integrals of this type; they fall within the general
framework of the Mellin transform, which associates the function
103
Theorem 13. Let k be a real number. The space H
1
k (P ) of holomorphic
functions such that
Im(z)>0
|f (z)| y
k−2 dxdy < +∞
(12.24)
does not reduce to {0} if and only if k > 1. Any holomorphic solution of (24)
is the complex Fourier transform of a continuous function ˆ
f (t) which is zero
for t < 0. Formulas (17), (18) and (19) hold.
Exercise 6. Calculate the function ˆ
f (t) corresponding to (23). p need
not be assumed to be an integer nor a real since function (24) has uniform
branches on y > 0.
To conclude this section, note that, while we have obtained important
properties of the functions ˆ
f , we have not characterized them as we did in
the Paley-Wiener theorem; as far as I know – given the flood of publications
since the last fifty years, it is necessary to be prudent –, the answer to this
question will never be known. It is, however, fully known if the functions f (z)
are taken to be square integrable on the half-plane: these are complex Fourier
transforms of the functions ˆ
f (t) which are zero for t ≤ 0 and for which
+∞
0
ˆ
f (t)
2
t
1−k dt < +∞ ;
but these are now functions on the Lebesgue L
2 space and the result cannot
be obtained without the help of the whole theory of Fourier transforms on
L
2 . This topic will be presented in Chap. XI.
13 – The Mellin Transform
(i) Questions of convergence. To obtain Paley-Wiener type theorems that hold
for meromorphic functions rather than holomorphic ones as in the previous
n
◦ – an important problem in analytic number theory for example –, it is
useful to reformulate the complex Fourier transform. If the change of variable
exp(2πt) = u is carried out in integral (12.1) which it is defined by, then u > 0
and
2πdt = du/u = d
∗ u .
Setting iz = s, we get e(tz) = u
s and ˆ
f (t) = F (e
2πt ), and so
2πf (z) =
+∞
0
F (u)u
s d
∗ u .
We often come across integrals of this type; they fall within the general
framework of the Mellin transform, which associates the function