100
VIII – Cauchy Theory
f
(r) (x + iy)
p
dx < +∞
for all y ∈]a, b[ and all r ∈ N. Converse ?
Returning to the case p = 1, this result sends us back to Chap. VII, n
◦ 29
where it was shown how to deduce Fourier’s inversion formula from Poisson’s
summation formula. Consider the function x → f (x + iy) for given y ∈ I. As
the series
f (z + n) converges normally on every compact subset,
60
|f (x + iy)| dx < +∞
(12.10)
for all y ∈]a, b[, which allows the Fourier transform
f (x + iy)e(−tx)dx = F (t, y)
(12.11)
to be defined; it is a continuous function of t.
Let us show that there is a function ˆ
f (t) on R such that
F (t, y) = ˆ
f (t)e(ity) .
(12.12)
Formula (11), multiplied by e(−ity) = exp(2πty), can also be written
F (t, y)e(−ity) =
f (z)e(−tz)dz .
This is the Cauchy integral along the horizontal Im(z) = y, and it all amounts
to showing that it is independent of y. To compare its values at y
and y
,
let us integrate over the rectangle ABCD bounded by the horizontals AB :
Im(z) = y
and DC : Im(z) = y
and the verticals DA : Re(z) = −n and BC :
Re(z) = n; by Cauchy, the result is zero. |f (z)e(−tz)| = |f (n + iy)| exp(2πty)
on BC; as the series
f (z + n) converges normally on the compact interval
[y
, y
] of the imaginary axis, the function y → f (n + iy) converges uniformly
to 0 on [y
, y
] as n increases. At the limit, the contributions from the vertical
sides are, therefore, zero. This gives the expected result. Hence, there is a
relation
f (x + iy)e(−tx)dx = ˆ
f (t)e(ity)
(12.13)
60 Applied to the double integral (6), the version of the Lebesgue-Fubini theorem
for lsc positive functions (Chap. V, n
◦ 33, theorem 31), only states that the
function y →
|f (x + iy)|dx, with values ≤ +∞ (large inequality), is integrable
as a lsc function, and so is finite“ almost everywhere ”; but it could very well be
infinite for some values of y. The fact that it is finite everywhere, with no zero
measure exception, is one of the many miracles of the theory of holomorphic
functions: all the seemingly pathological phenomena of the theory of integration
disappear. This meta-theorem useful for making conjectures on what is going on,
does not exempt from giving proper proofs.
VIII – Cauchy Theory
f
(r) (x + iy)
p
dx < +∞
for all y ∈]a, b[ and all r ∈ N. Converse ?
Returning to the case p = 1, this result sends us back to Chap. VII, n
◦ 29
where it was shown how to deduce Fourier’s inversion formula from Poisson’s
summation formula. Consider the function x → f (x + iy) for given y ∈ I. As
the series
f (z + n) converges normally on every compact subset,
60
|f (x + iy)| dx < +∞
(12.10)
for all y ∈]a, b[, which allows the Fourier transform
f (x + iy)e(−tx)dx = F (t, y)
(12.11)
to be defined; it is a continuous function of t.
Let us show that there is a function ˆ
f (t) on R such that
F (t, y) = ˆ
f (t)e(ity) .
(12.12)
Formula (11), multiplied by e(−ity) = exp(2πty), can also be written
F (t, y)e(−ity) =
f (z)e(−tz)dz .
This is the Cauchy integral along the horizontal Im(z) = y, and it all amounts
to showing that it is independent of y. To compare its values at y
and y
,
let us integrate over the rectangle ABCD bounded by the horizontals AB :
Im(z) = y
and DC : Im(z) = y
and the verticals DA : Re(z) = −n and BC :
Re(z) = n; by Cauchy, the result is zero. |f (z)e(−tz)| = |f (n + iy)| exp(2πty)
on BC; as the series
f (z + n) converges normally on the compact interval
[y
, y
] of the imaginary axis, the function y → f (n + iy) converges uniformly
to 0 on [y
, y
] as n increases. At the limit, the contributions from the vertical
sides are, therefore, zero. This gives the expected result. Hence, there is a
relation
f (x + iy)e(−tx)dx = ˆ
f (t)e(ity)
(12.13)
60 Applied to the double integral (6), the version of the Lebesgue-Fubini theorem
for lsc positive functions (Chap. V, n
◦ 33, theorem 31), only states that the
function y →
|f (x + iy)|dx, with values ≤ +∞ (large inequality), is integrable
as a lsc function, and so is finite“ almost everywhere ”; but it could very well be
infinite for some values of y. The fact that it is finite everywhere, with no zero
measure exception, is one of the many miracles of the theory of holomorphic
functions: all the seemingly pathological phenomena of the theory of integration
disappear. This meta-theorem useful for making conjectures on what is going on,
does not exempt from giving proper proofs.