(February 14, 2017)
The Hilbert transform
Paul Garrett [email protected] http://www.math.umn.edu/egarrett/
[This document is http://www.math.umn.edu/egarrett/m/fun/notes 2016-17/hilbert transform.pdf] 1. The principal-value functional 2. Other descriptions of the principal-value functional 3. Uniqueness of odd/even homogeneous distributions 4. Hilbert transforms 5. Examples 6. ... 7. Appendix: uniqueness of equivariant functionals 8. Appendix: distributions supported at {0} 9. Appendix: the snake lemma
Formulaically, the Cauchy principal-value functional η is
- Z
- Z
∞
- f(x)
- f(x)
- ηf = principal-value functional of f = P.V.
- dx = lim
dx x
ε→0+
x
−∞
|x|>ε
This is a somewhat fragile presentation. In particular, the apparent integral is not a literal integral! The uniqueness proven below helps prove plausible properties like the Sokhotski-Plemelj theorem from [Sokhotski 1871], [Plemelji 1908], with a possibly unexpected leading term:
- Z
- Z
- ∞
- ∞
- f(x)
- f(x)
lim
- dx = −iπ f(0) + P.V.
- dx
ε→0+
- x + iε
- x
- −∞
- −∞
See also [Gelfand-Silov 1964]. Other plausible identities are best certified using uniqueness, such as
Z
f(x) − f(−x)
12
- ηf =
- dx
(for f ∈ C1(R) ∩ L1(R))
x
R
f(x)−f(−x)
- using the canonical continuous extension of
- at 0. Also,
x
Z
2
f(x) − f(0) · e−x
- ηf =
- dx
(for f ∈ Co(R) ∩ L1(R))
x
R
The Hilbert transform of a function f on R is awkwardly described as a principal-value integral
- Z
- Z
∞
1
(Hf)(x) = P.V.
π
- f(t)
- 1
- f(t)
dt =
lim
dt x − t π ε→0+ x − t
|t−x|>ε
−∞
with the leading constant 1/π understandable with sufficient hindsight: we will see that this adjustment makes H extend to a unitary operator on L2(R). The formulaic presentation of H makes it appear to be a convolution with (a constant multiple of) the principal-value functional. The fragility of these presentations make existence, continuity properties, and range of applicability of H less clear than one would like.
1
Paul Garrett: The Hilbert transform (February 14, 2017)
1. The principal-value functional
The principal-value functional η is better characterized as the unique (up to a constant multiple) odd distribution on R, positive-homogeneous of degree 0 as a distribution (see below). This characterization allows unambiguous comparison of various limiting expressions closely related to the principal-value functional. Further, the characterization of the principal-value functional makes discussion of the Hilbert transform less fragile and more convincing.
Again, let
- Z
- Z
∞
- f(x)
- f(x)
- ηf = principal-value functional of f = P.V.
- dx = lim
dx x
ε→0+
x
−∞
|x|>ε
We prove that η is a tempered distribution: [1.1] Claim: For f ∈ S , for every h > 0,
|η(f)| ≤
Z
ꢀ
ꢀꢀ
ꢀꢀꢀ
f(x) dx + 2h sup |f0(x)|
x
- |x|≤h
- |x|≥h
Thus, u is a tempered distribution:
- |η(f)| ꢀ sup(1 + x2)|f(x)| + sup |f0(x)|
- (implied constant independent of f)
- x
- x
Proof: Certainly
- Z
- Z
ꢀ
ꢀꢀ
ꢀꢀꢀ
ꢀꢀꢀ
ꢀꢀꢀ
- f(x)
- f(x)
- |η(f)| =
- dx + lim
dx x
ε→0+
x
|x|≥h
ε<|x|≤h
By the Mean Value Theorem, f(x) = f(0) + x f0(ξx) for some ξx between 0 and x. Thus,
- Z
- Z
- Z
- Z
- f(x)
- f(0)
- dx =
- dx +
f(ξx) dx = 0 +
f(ξx) dx
- x
- x
- ε<|x|≤h
- ε<|x|≤h
- ε<|x|≤h
- ε<|x|≤h
because 1/x is odd and x → f(0) is even. Thus,
- Z
- Z
- Z
ꢀ
ꢀꢀ
ꢀꢀꢀ
ꢀꢀꢀ
ꢀꢀꢀ
f(x)
dx
=
f(ξx) dx
≤
|f(ξx)| dx x
- ε<|x|≤h
- ε<|x|≤h
- ε<|x|≤h
≤ 2h · sup |f0(x)| ≤ 2h · sup |f0(x)|
ε<|x|≤h
|x|≤h
The latter is independent of ε. With 0 < h ≤ 1,
- Z
- Z
- Z
ꢀ
ꢀꢀ
ꢀꢀꢀ
ꢀꢀꢀ
ꢀꢀꢀ
ꢀꢀꢀ
ꢀ
- f(x)
- f(x)
- f(x)
ꢀ
- dx =
- dx +
- dx
ꢀ
- x
- x
- x
- |x|≥h
- |x|≥1
- h≤|x|≤1
- Z
- Z
- Z
- 1
- 1
≤
|f(x)| dx +
|f(x)| dx ≤
- (1 + x2)|f(x)| ·
- dx + 2h sup |f(x)|
h
1 + x2
x
- |x|≥1
- h≤|x|≤1
- |x|≥1
Z
1
- ≤ sup(1 + x2)|f(x)| ·
- dx + 2h sup |f(x)| ꢀ sup(1 + x2)|f(x)|
1 + x2
- x
- x
- x
|x|≥1
Of course, sup|x|≤h |f0(x)| ≤ supx |f0(x)|, giving the corollary.
///
2
Paul Garrett: The Hilbert transform (February 14, 2017)
To make the degree of a positive-homogeneous function ϕ agree with the degree of the integration-against-ϕ distribution uϕ, due to change-of-measure, we find that
- Z
- Z
(uϕ ◦ t)(f) = uϕ(f ◦ t−1) =
uϕ(x) f(t−1x) dx =
uϕ(tx) f(x) d(tx) = t · uϕ◦t(f)
- R
- R
Thus, for agreement of the notion of homogeneity for distributions and (integrate-against) functions, the dilation action u → u ◦ t on distributions should be
1
(u ◦ t)(f) = u(f ◦ t−1
- )
- (for t > 0, test function f, and distribution u)
t
Parity is as expected: u is odd when u(x → f(−x)) = −u(x → f(x)) for all test functions f, and is even when u(x → f(−x)) = +u(x → f(x)) for all test functions f.
[1.2] Claim: The principal-value distribution η is positive-homogeneous of degree −1, and is odd. Proof: The εth integral in the limit definition of η is itself odd, by changing variables:
- Z
- Z
- f(−x)
- f(x)
η(f ◦ (−1)) = lim dx = lim d(−x)
ε→0+
x
ε→0+
−x
- |x|≥ε
- |x|≥ε
- Z
- Z
- f(x)
- f(x)
=
- lim −
- dx = − lim
dx = −η(f)
ε→0+
x
ε→0+
x
- |x|≥ε
- |x|≥ε
as claimed. For the degree of homogeneity, for t > 0 and test function f,
- Z
- Z
- Z
- f(xt )
- 1
- 1
- 1
- 1
- f(x)
- 1
- f(x)
(η ◦t)(f) = η(f ◦ ) = lim
dx =
Z
- lim
- d(tx) =
- lim
- d(x)
- t
- t
- t ε→0+
- x
- t ε→0+
- tx
- t ε→0+
- x
|x|≥ε
- |tx|≥ε
- |x|≥ε/t
- 1
- f(x)
- 1
- =
- lim
- d(x) = η(f)
- t ε→0+
- x
- t
|x|≥ε
That is, η is homogeneous of degree −1.
///
2. Other descriptions of the principal-value functional
[2.1] Claim: Let ϕ be any even function in L1(R) ∩ C1(R), with ϕ(0) = 1. Then
- Z
- Z
- f(x)
- f(x) − f(0) · ϕ(x)
- P.V.
- dx =
- dx
(for f ∈ Co(R) ∩ L1(R), for example)
- x
- x
- R
- R
- ꢁ
- ꢂ
where f(x) − f(0) · ϕ(x) /x is extended by continuity at x = 0. Proof: By the demonstrated odd-ness of the principal-value integral, it is 0 on the even function ϕ.
f(x)−f(0)·ϕ(x)
- Extending
- by continuity at 0,
x
- Z
- Z
- f(x) − f(0) · ϕ(x)
- f(x) − f(0) · ϕ(x)
dx = lim
dx x
ε→0+
x
R
|x|≥ε
- Z
- Z
- Z
- f(x)
- ϕ(x)
- f(x)
- =
- lim
- dx + f(0) lim
dx = P.V.
dx + 0
ε→0+
x
ε→0+
- x
- x
- |x|≥ε
- |x|≥ε
R
as claimed.
///
3
Paul Garrett: The Hilbert transform (February 14, 2017)
The leading term in the following might be unexpected:
[2.2] Claim: (Sokhotski-Plemelj)
- Z
- Z
- ∞
- ∞
- f(x)
- f(x)
lim
- dx = −iπ f(0) + P.V.
- dx
ε→0+
- x + iε
- x
- −∞
- −∞
Proof: Aiming to apply the previous claim,
- Z
- Z
- Z
f(0) 1+x2
f(x) −
- f(x)
- 1/(1 + x2)
- lim
- dx = lim
- dx + f(0) lim
dx
ε→0+
x + iε
ε→0+
x + iε
ε→0+
x + iε
- R
- R
- R
For 0 < ε < 1, the right-most integral is evaluated by residues, moving the contour through the upper half-plane:
Z
- 1/(1 + x2)
- 1
- 1
πidx = 2πi Resx=i
= 2πi
−→
= −iπ x + iε
- (1 + x2)(x + iε)
- (i + i)(i + iε)
R
as claimed. [2.3] Claim: With f continuously differentiable at 0, thereby extending
///
f(x)−f(−x)
by continuity at 0,
x
- Z
- Z
∞
- f(x) − f(−x)
- f(x)
12
- dx = P.V.
- dx
(for f ∈ C1(R) ∩ L1(R))
- x
- x
R
−∞
f(x)−f(−x)
Proof: For test function f, since functional, is continuous as 0, using the odd-ness of the principal-value
x
- Z
- Z
- Z
- f(x) − f(−x)
- f(x) − f(−x)
f(x)
12
12
dx =
- lim
- dx = lim
dx x
R
ε→0+
x
ε→0+
x
- |x|≥ε
- |x|≥ε
as claimed.
///
3. Uniqueness of odd/even homogeneous distributions
After some clarification of notions of parity and positive-homogeneity, we show that there is a unique distribution of a given parity and positive-homogeneity degree.
d
[3.1] Claim: For a positive-homogeneous distribution u of degree s, xdx u is again positive-homogeneous of degree s, and with the same parity.
Proof: For test function f,
d
1
d
1
d
1
d
- ((x u) ◦ t)(f) = (x u)(f ◦ t−1) =
- (
- u)(x · f ◦ t−1) = − u( (x · (f ◦ t−1)))
- dx
- t
- dx
- t dx
- t
- dx
- 1
- 1
- 1
- = − u(f ◦ t−1 + x · t−1 · (f0 ◦ t−1)) = − u(f ◦ t−1 + (x · f0) ◦ t−1)) = − u((f + xf0) ◦ t−1
- )
- t
- t
- t
- ꢁ
- ꢂ
- d
- d
- d
= −(u ◦ t)(f + xf0) = −(u ◦ t)( (xf)) = ts · − u( (xf)) = ts · (x u)(f)
- dx
- dx
- dx
4
Paul Garrett: The Hilbert transform (February 14, 2017)
as asserted. Preservation of parity is similar: writing f−(x) = f(−x) for functions f, and u−(f) = u(f−) for distributions u,
- d
- d
- d
- d
- d
(x u)−(f) = (x u)(f−) = ( u)(x · f−) = −u( (x · f−)) = u( ((x · f)−))
- dx
- dx
- dx
- dx
- dx
ꢁ
ꢂ−
- d
- d
- d
- = u(−
- (x · f) ) = −u−( (x · f)) = ((x )u−)(f)
- dx
- dx
- dx
as claimed.
///
d
The Euler operator xdx is the infinitesimal form of dilation f(x) → f(tx). This is manifest in the following: