92
VIII – Cauchy Theory
This is the Cauchy principal value for integrable functions on R and differentiable at the origin, which an essential condition. We thus finally get the
formula
p.v.
f (t)dt/t = −πi
sgn(u) ˆ
f (u)du ,
(11.15)
which at least holds under the following conditions: f is differentiable at the
origin and belongs to F
1 (R).
It is in particular the case if f ∈ S(R), the Schwartz space. As a function
of f , the left hand side T (f ) can immediately be checked to be a tempered
distribution (Chapter VII, § 6, n
◦ 32), denoted v.p.1/t by some; indeed,
|T (f )| ≤ π
ˆ
f (u)
du = π
ˆ
f (u)
1 + 4π
2 u
2
.
1 + 4π
2 u
2
−1 du ≤
≤ sup
ˆ
f (u)
1 + 4π
2 u
2
.
Now, (1 + 4π
2 u
2 ) ˆ
f (u) is the Fourier transform of f (t) − f
(t), and hence for
all u is bounded above by the integral
|f (t) − f
(t)| dt =
1 + t
2
|f (t) − f
(t)| .
1 + t
2
−1 dt ≤
≤ π. sup
1 + t
2
|f (t) − f
(t)| ≤
≤ π. sup
1 + t
2
|f (t)| + π. sup
1 + t
2
|f
(t)| ≤ N 2 (f )
up to a constant factor, and where, as in Chapter VII, § 6, eq. (32.3), the
topology of S(R) is defined by the seminorms
N r (f ) =
p,q ≤ r
sup
t
p f
(q) (t)
.
This proves that f → T (f ) is continuous with respect to the topology of
S(R).
The Fourier transform of the tempered distribution T is, by definition, the
distribution ˆ
T (f ) = T ( ˆ
f ); therefore, the significance of the formula obtained
is that the Fourier transform of the distribution v.p. 1/t is the distribution
−πi.sgn(u), which is in fact a function since, in distribution theory, a function
ϕ(u) is always identified with the distribution f →
f (u)ϕ(u)du. Conversely,
the Fourier transform of the function −πi.sgn(u), a Fourier transform that is
not well-defined in classical theory as the function sgn(u) is not integrable,
is v.p. 1/t. This theory has given rise to generalizations in several dimensions
that play an important role in some aspects of the theory of partial differential
equations.
VIII – Cauchy Theory
This is the Cauchy principal value for integrable functions on R and differentiable at the origin, which an essential condition. We thus finally get the
formula
p.v.
f (t)dt/t = −πi
sgn(u) ˆ
f (u)du ,
(11.15)
which at least holds under the following conditions: f is differentiable at the
origin and belongs to F
1 (R).
It is in particular the case if f ∈ S(R), the Schwartz space. As a function
of f , the left hand side T (f ) can immediately be checked to be a tempered
distribution (Chapter VII, § 6, n
◦ 32), denoted v.p.1/t by some; indeed,
|T (f )| ≤ π
ˆ
f (u)
du = π
ˆ
f (u)
1 + 4π
2 u
2
.
1 + 4π
2 u
2
−1 du ≤
≤ sup
ˆ
f (u)
1 + 4π
2 u
2
.
Now, (1 + 4π
2 u
2 ) ˆ
f (u) is the Fourier transform of f (t) − f
(t), and hence for
all u is bounded above by the integral
|f (t) − f
(t)| dt =
1 + t
2
|f (t) − f
(t)| .
1 + t
2
−1 dt ≤
≤ π. sup
1 + t
2
|f (t) − f
(t)| ≤
≤ π. sup
1 + t
2
|f (t)| + π. sup
1 + t
2
|f
(t)| ≤ N 2 (f )
up to a constant factor, and where, as in Chapter VII, § 6, eq. (32.3), the
topology of S(R) is defined by the seminorms
N r (f ) =
p,q ≤ r
sup
t
p f
(q) (t)
.
This proves that f → T (f ) is continuous with respect to the topology of
S(R).
The Fourier transform of the tempered distribution T is, by definition, the
distribution ˆ
T (f ) = T ( ˆ
f ); therefore, the significance of the formula obtained
is that the Fourier transform of the distribution v.p. 1/t is the distribution
−πi.sgn(u), which is in fact a function since, in distribution theory, a function
ϕ(u) is always identified with the distribution f →
f (u)ϕ(u)du. Conversely,
the Fourier transform of the function −πi.sgn(u), a Fourier transform that is
not well-defined in classical theory as the function sgn(u) is not integrable,
is v.p. 1/t. This theory has given rise to generalizations in several dimensions
that play an important role in some aspects of the theory of partial differential
equations.