The Hilbert Transform

(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 ∞ f(x) Z f(x) ηf = principal-value functional of f = P.V. dx = lim dx −∞ x ε→0+ |x|>ε x

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 ∞ f(x) Z ∞ 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) ηf = 1 dx (for f ∈ C1( ) ∩ L1( )) 2 x R R R

f(x)−f(−x) using the canonical continuous extension of x at 0. Also,

2 Z f(x) − f(0) · e−x ηf = dx (for f ∈ Co( ) ∩ L1( )) x R R R

The Hilbert transform of a function f on R is awkwardly described as a principal-value integral

1 Z ∞ f(t) 1 Z f(t) (Hf)(x) = P.V. dt = lim dt π −∞ x − t π ε→0+ |t−x|>ε x − t 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 ∞ f(x) Z f(x) ηf = principal-value functional of f = P.V. dx = lim dx −∞ x ε→0+ |x|>ε x

We prove that η is a tempered distribution:

[1.1] Claim: For f ∈ S , for every h > 0,

Z f(x) 0 |η(f)| ≤ dx + 2h sup |f (x)| |x|≥h x |x|≤h

Thus, u is a tempered distribution:

|η(f)| sup(1 + x2)|f(x)| + sup |f 0(x)| (implied constant independent of f) x x

Proof: Certainly Z f(x) Z f(x) |η(f)| = dx + lim dx + |x|≥h x ε→0 ε<|x|≤h x 0 By the Mean Value Theorem, f(x) = f(0) + x f (ξx) for some ξx between 0 and x. Thus,

Z f(x) Z f(0) Z Z dx = dx + f(ξx) dx = 0 + f(ξx) dx ε<|x|≤h x ε<|x|≤h x ε<|x|≤h ε<|x|≤h because 1/x is odd and x → f(0) is even. Thus,

Z f(x) Z Z

dx = f(ξx) dx ≤ |f(ξx)| dx ε<|x|≤h x ε<|x|≤h ε<|x|≤h

≤ 2h · sup |f 0(x)| ≤ 2h · sup |f 0(x)| ε<|x|≤h |x|≤h The latter is independent of ε. With 0 < h ≤ 1, Z f(x) Z f(x) Z f(x)

dx = dx + dx |x|≥h x |x|≥1 x h≤|x|≤1 x Z Z Z 1 2 1 ≤ |f(x)| dx + |f(x)| dx ≤ (1 + x )|f(x)| · 2 dx + 2h sup |f(x)| |x|≥1 h h≤|x|≤1 |x|≥1 1 + x x Z 2 1 2 ≤ sup(1 + x )|f(x)| · 2 dx + 2h sup |f(x)| sup(1 + x )|f(x)| x |x|≥1 1 + x x x

0 0 Of course, sup|x|≤h |f (x)| ≤ supx |f (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 −1 −1 (uϕ ◦ t)(f) = uϕ(f ◦ t ) = uϕ(x) f(t x) 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 f(−x) Z f(x) η(f ◦ (−1)) = lim dx = lim d(−x) ε→0+ |x|≥ε x ε→0+ |x|≥ε −x

Z f(x) Z f(x) = lim − dx = − lim dx = −η(f) ε→0+ |x|≥ε x ε→0+ |x|≥ε x as claimed. For the degree of homogeneity, for t > 0 and test function f,

1 1 1 Z f( x ) 1 Z f(x) 1 Z f(x) (η ◦ t)(f) = η(f ◦ ) = lim t dx = lim d(tx) = lim d(x) t t t ε→0+ |x|≥ε x t ε→0+ |tx|≥ε tx t ε→0+ |x|≥ε/t x

1 Z f(x) 1 = lim d(x) = η(f) t ε→0+ |x|≥ε x t 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 f(x) Z f(x) − f(0) · ϕ(x) P.V. dx = dx (for f ∈ Co( ) ∩ L1( ), for example) x x R R R R 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 x by continuity at 0,

Z f(x) − f(0) · ϕ(x) Z f(x) − f(0) · ϕ(x) dx = lim dx x ε→0+ x R |x|≥ε Z f(x) Z ϕ(x) Z 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 ∞ f(x) Z ∞ f(x) lim dx = −iπ f(0) + P.V. dx ε→0+ −∞ x + iε −∞ x

Proof: Aiming to apply the previous claim,

f(0) 2 Z f(x) Z f(x) − 2 Z 1/(1 + x ) lim dx = lim 1+x 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 π dx = 2πi Res = 2πi −→ = −iπ x + iε x=i (1 + x2)(x + iε) (i + i)(i + iε) i R as claimed. ///

f(x)−f(−x) [2.3] Claim: With f continuously diﬀerentiable at 0, thereby extending x by continuity at 0,

Z f(x) − f(−x) Z ∞ f(x) 1 dx = P.V. dx (for f ∈ C1( ) ∩ L1( )) 2 x x R R R −∞

f(x)−f(−x) Proof: For test function f, since x is continuous as 0, using the odd-ness of the principal-value functional, Z Z Z 1 f(x) − f(−x) 1 f(x) − f(−x) f(x) 2 dx = 2 lim dx = lim dx x ε→0+ x ε→0+ x R |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, x dx 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 · (f 0 ◦ t−1)) = − u(f ◦ t−1 + (x · f 0) ◦ t−1)) = − u((f + xf 0) ◦ t−1) t t t d d d = −(u ◦ t)(f + xf 0) = −(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

d − d d = u(− (x · f) ) = −u−( (x · f)) = ((x )u−)(f) dx dx dx as claimed. ///

d The Euler operator x dx is the inﬁnitesimal form of dilation f(x) → f(tx). This is manifest in the following:

d [3.2] Corollary: x dx u = s · u for positive-homogeneous u of degree s.

Proof: Diﬀerentiate the distribution-valued equality u ◦ t = ts · u with respect to t and set t = 1: the right- hand side is s · u. For the left-hand side, for test function f, since t → u ◦ t is a smooth distribution-valued function of t,

∂ ∂ ∂ 1 1 ∂ 1 1 −1 1 x 1 ( (u ◦ t))(f) = ((u ◦ t)(f)) = ( u(f ◦ )) = u( ( f ◦ )) = u (f ◦ ) − · (f 0 ◦ ) ∂t ∂t ∂t t t ∂t t t t2 t t2 t

Evaluating at t = 1 gives

d (s · u)(f) = u − f − x · f 0 = −u(f) − (x · u)(f 0) = −u(f) + (x · u) (f) dx

d = −u(f) + (1 · u + x · u0)(f) = x u (f) dx as claimed. ///

[3.3] Theorem: Up to constant multiples, the principal-value distribution η is the unique odd distribution of positive-homogeneity degree −1. More generally, except for s = −n with ε = (−1)n+1 with n = 1, 2,..., there is a unique (up to constant multiples) distribution of parity ε and of positive-homogeneity degree s.

Proof: Fix parity ε = ±, and let V be the set of test functions of parity ε vanishing to inﬁnite order at 0. This is the closure, in the space Dε = Dε(R) of test functions of parity ε, of the space of test functions on ∞ R with support not containing 0 and of parity ε. On Cc (0, +∞) there is a unique continuous functional ∞ on Cc (0, +∞) positive homogeneous of degree s (as distribution), by the appendix, given by integration s ε against |x| . Given a choice of parity ε, there is a unique extension to a continuous linear functional vs on ε Do, and of positive-homogeneity degree s. Dualize the short exact sequence

0 −→ V −→ Dε −→ Dε/V −→ 0 to a short exact sequence (invoking Hahn-Banach for exactness at the right-most joint)

∗ ∗ ∗ 0 −→ (Dε/V ) −→ Dε −→ V −→ 0

The quotient Dε/V is identiﬁable with germs of smooth functions of parity ε supported at 0 modulo germs of smooth functions vanishing identically at 0 of parity ε. The dual is the collection of distributions supported at 0 of parity ε, which by the theory of Taylor-Maclaurin series (see appendix) consists of ﬁnite linear combinations of Dirac δ and its derivatives, or parity ε. Of course, δ is even, δ0 is odd, and so on.

5 Paul Garrett: The Hilbert transform (February 14, 2017)

d Let T = x dx − s, and consider the very small (vertical) complex of short exact sequences

0 0 0

∗ ∗ ∗ 0 / (Dε/V ) / Dε / V / 0

T T T

∗ ∗ ∗ 0 / (Dε/V ) / Dε / V / 0

0 0 0

The corresponding long (co-) homology sequence is

(D /V )∗ D∗ V ∗ 0 → ker T → ker T → ker T → ε → ε → → 0 ∗ ∗ ∗ ∗ ∗ ∗ (Dε/V ) Dε V T ((Dε/V ) ) T (Dε ) T (V )

ε The functional v is in ker T | ∗ , and we hope to extend it uniquely to a functional in ker T | ∗ . It would suﬃce s V Dε ∗ ∗ ∗ that ker T(Dε/V ) = 0 and T ((Dε/V ) ) = (Dε/V ) , that is, T is an isomorphism on distributions supported on {0} of parity ε and positive-homogeneity degree s. The only failures are s = −n with ε = (−1)n+1, because d x − (−n) δ(n) = 0 (for n = 1, 2,...) dx For the principal-value functional, although s = −1 does occur as a pole for parity +1, the parity is ε = −1, so there is no obstruction to the extension, and it is unique. ///

ε s s For Re(s) > −1, let us be the distributions given by integration against |x| for ε = +1 and sgn x · |x| for ε = −1. For Re(s) > −1, these are locally integrable, and of respective parities ε = ±1 and of positive- homogeneity degree s.

ε [3.4] Corollary: The distributions us have meromorphic continuations to tempered-distribution-valued functions of s ∈ C, with only simple poles. For ε = +1, the poles are at −n = −1, −3, −5,..., with residues constant multiples of δ(n). For ε = −1, the poles are at −n = −2, −4, −6,..., with residues constant multiples of δ(n). The meromorphic continuations maintain the positive-homogeneity and parity. ε 1 d −ε For s 6= −1, us = s+1 dx us+1. For s = −1, ∂ ∂ ∂ −1 +1 −1 u0 = 2 · δ and us = u−1 = η ∂x ∂x ∂s s=0

ε Proof: The functionals vs on V are entire, since there is no convergence issue for test functions vanishing to ε inﬁnite order at 0. Thus, at pole −n, lims→−n(s + n)us vanishes on test functions vanishing to inﬁnite order ε at 0. Thus, the residue lims→−n(s + n)us is a distribution supported on {0}. It has the same parity ε, and is positive-homogeneous of degree −n, in the distributional sense. Thus, there is no pole unless ε = (−1)n, for n = 0, −1, −2,....

ε s s For Re(s) 1, the integrate-against functionals us given by |x| for ε = +1 and sgnx · |x| for ε = −1 are of ε (distributional) positive-homogeneity degree s, and parity ε. For each Re(s) 1, us is the unique extensions ε of the distributions vs, up to constants possibly depending on s. By the vector-valued form of the identity principle from complex analysis, the same homogeneity and parity properties hold for any meromorphic d ε ε continuation. From x dx us = s · us, d 1 u−ε = · s · u−ε = s · uε dx s x s s−1 6 Paul Garrett: The Hilbert transform (February 14, 2017)

ε 1 d −ε −ε Thus, us = s+1 dx us+1. Since us+1 is a holomorphic distribution-valued function in Re(s) > −2, this extends ε ± us to Re(s) > −2, except for possible pole at s = −1. By induction, us has a meromorphic continuation to all of C except for possible (simple) poles at negative integers. As above, the only possible residues are distributions supported at 0, which can only occur at non-positive even integers, or non-positive odd integers, depending on parity.

d −ε ε +1 For ε = +1, the relation dx us = s · us−1 evaluated at s = 0 yields the residue of us at −1, namely, a constant multiple of δ. For ε = −1, there is no pole, and evaluation of the relation at s = 0 just gives d ∂ −ε ε dx 1 = 0. Thus, we diﬀerentiate in s before evaluation: the relation ∂x us = s · us−1 gives ∂ ∂ ∂ ∂ ∂ ∂ +1 +1 −1 −1 −1 −1 us = us = s · us−1 = (us−1 + s · us ) = u−1 ∂x ∂s s=0 ∂s s=0 ∂x ∂s s=0 ∂s s=0 Of course, ∂ +1 ∂ s s us = |x| = (|x| · log |x|) = log |x| ∂s s=0 ∂s s=0 s=0 This completes the discussion. ///

[3.5] Remark: In particular, the meromorphic continuations are tempered distributions. That is, all positive- homogeneous distributions are tempered. Also, away from 0, homogeneous distributions are given locally by smooth functions. 4. Hilbert transforms

The most immediate description of the Hilbert transform Hf is 1 Z f(y) 1 Z f(y) (Hf)(y) = P.V. dy = lim dy π x − y π ε→0+ x − y R |x|≥ε From the previous discussion of the principal-value functional η, Hf exists at least as a point-wise function. ∞ In fact, with (Ly)f(x) = f(x − y), y → Lyf is a smooth S -valued function, so f → η(Lyf) is in C (R), but growth properties are unclear. The usual heuristic is Z f(y) P.V. dy = (η ∗ f)(x) x − y R so 1 Hf = η ∗ f (for f ∈ S ( )) π R Granting this for a moment, taking Fourier transform would seem to give 1 (Hf) = η · fb b π b The principal-value functional η is a tempered distribution, so its Fourier transform makes sense at least as a tempered distribution.

[4.1] Claim: The Fourier transform of an odd/even distribution of positive-homogeneity degree s is odd/even of positive-homogeneity degree −(s + 1).

Proof: Since Schwartz functions are dense, the interaction of dilation and Fourier transform can be examined via functions with point-wise values and Fourier transforms deﬁned by literal integrals, although none of these can be homogeneous. It is immediate by changing variables that parity is preserved. With (f ◦t)(x) = f(tx), for functions f, Z 1 Z 1 1 (f ◦ t) (ξ) = e−2πiξxf(tx) dx = e−2πiξx/tf(tx) dx = (fb◦ )(ξ) b t t t R R 7 Paul Garrett: The Hilbert transform (February 14, 2017)

That is, 1 1 fb◦ t = · f ◦ t t Thus, without trying to use pointwise sense of a tempered distribution, 1 1 1 1 1 1 1 (u ◦ t)(f) = u( f ◦ ) = u (f ◦ ) = u t · fb◦ t = (u ◦ )(fb) b b t t t t b t t t

11s −(s+1) −(s+1) = · u (fb) = t · u(fb) = t · u(f) t t b as claimed. ///

[4.2] Corollary: The principal-value distribution η, has Fourier transform has degree 0 and of odd parity. In particular, ηb = −iπ sgn x. Proof: By uniqueness of distributions of a given degree and parity, it suﬃces to evaluate η and sgnx on a 2 b given odd Schwartz function, for example, x → xe−πx . On one hand,

−πx2 2 2 2 Z xe Z 2 η(xe−πx ) = η (xe−πx ) = η i−1xe−πx ) = i−1 dx = i−1 e−πx dx = i−1 b b x R R On the other,

∞ ∞ ∞ Z 2 Z 2 Z 2 Z 2 dx Z dx sgn x · xe−πx dx = |x| e−πx dx = 2 x e−πx dx = 2 x2 e−πx = x e−πx x x R R 0 0 0 Z ∞ dx = π−1 x e−x = π−1Γ(1) = π−1 0 x Thus, ηb = −iπ sgn x. /// With the factors of π cancelling, this suggests that we could prove

∨ [4.3] Corollary: Hf = − i sgn x · fb [... iou ...]

[4.4] Corollary: The Hilbert transform extends by continuity to an isometric isomorphism of L2 to itself.

Proof: Assuming the corollary, since Fourier tranform is an isometry of L2 to itself, and multiplication by sgn is an isometry, the Hilbert transform is an L2-isometry on S ∩ L2, and extends by continuity to L2. ///

[4.5] Corollary: H ◦ Hf(x) = f(−x) for f ∈ S or f ∈ L2. ///

But what meaning should convolution of f ∈ S with η ∈ S ∗ have? [iou] 5. Examples

∨ ∨ ∨ δ1 − δ−1 δ1 + δ−1 [5.1] Example: H sin x = − i sgn x · sinc = − i sgn x · = − = − cos x 2i 2

sin 2πξ [5.2] Example: Since chd[−1,1] = πξ ,

sin 2πx ∨ ∨ e−2πiξ − 1 − 1 + e2πiξ cos 2πξ − 1 H = − i sgn x · ch = i ch − ch ) = i = πx [−1,1] [−1,0] [0,1] 2πiξ πξ

8 Paul Garrett: The Hilbert transform (February 14, 2017)

[5.3] Example: H|x|s [... iou ...]

6. ...

[... iou ...]

7. Appendix: uniqueness of equivariant functionals

o [7.1] Theorem: Let G be a topological group. Let V ⊂ Cc (G) be a quasi-complete locally convex topological vector space of complex-valued functions on G stable under left and right translations, and containing an approximate identity {ϕi}. Then there is a unique right G-invariant continuous functional on V (up to constant multiples): it is Z f → f(g) dg (with right translation-invariant measure) G

Proof: Let Tgf(y) = f(yg) be right translation. With right translation-invariant measure dg on G, since R G ϕi(g) dg = 1 and ϕi is non-negative, ϕi(g) dg is a probability measure on the (compact) support of ϕi. Thus, for any f ∈ V , we have a V -valued Gelfand-Pettis integral Z

Tϕi f = ϕi(g) Tgf dg ∈ closure of convex hull of {ϕi(g)f : g ∈ G} ⊂ V G o The assumption V ⊂ Cc (G) is meant to entail that the translation action of G on V is continuous, meaning that G × V → V by g × f → Tgf is (jointly) continuous. By continuity, given a neighborhood N of 0 in

V , we have Tϕi f ∈ f + N for all suﬃciently large i. That is, Tϕi f → f. For a right-invariant (continuous) functional u on V , Z u(f) = lim u g → ϕi(h) f(gh) dh i G This is Z Z −1 −1 u g → f(hg) ϕi(h ) dh = u g → f(h) ϕi(gh ) dh G G by replacing h by hg−1. By properties of Gelfand-Pettis integrals, and since f is guaranteed to be a compactly-supported continuous function, we can move the functional u inside the integral: the above becomes Z −1 f(h) u g → ϕi(gh ) dh G Using the right G-invariance of u the evaluation of u with right translation by h−1 gives Z Z f(h) u(g → ϕi(g)) dh = u(ϕi) · f(h) dh G G By assumption the latter expressions approach u(f) as i → ∞. For f so that the latter integral is non-zero, we see that the limit of the u(ϕi) exists, and then we conclude that u(f) is a constant multiple of the indicated integral with right Haar measure. ///

[7.2] Corollary: In the situation of the theorem, for a continuous group homomorphism χ : G → C×, suppose that V is stable under multiplication by the function χ. Then there is a unique continuous functional λ on V such that λ(Tgf) = χ(g) · λ(f) for all g ∈ G.

Proof: Let u be the G-invariant functional of the theorem, and put λ(f) = u(χ · f). ///

9 Paul Garrett: The Hilbert transform (February 14, 2017)

8. Appendix: distributions supported at {0}

[8.1] Theorem: A distribution u with support {0} is a (ﬁnite) linear combination of Dirac’s δ and its derivatives.

Recall: the support of a distribution u is the complement of the union of all open sets U ∈ Rn such that

u(f) = 0 (for f ∈ DK with compact K ⊂ U)

n ∗ Proof: Since the space D of test functions on R is D = colimK DK , it suﬃces to classify u in DK with support {0}.

A continuous linear map T from a limit of Banach spaces (such as DK ) to a normed space (such as C) factors through a limitand, under the mild assumption that the images of the limit in the limitands are dense. Thus, there is an order k ≥ 0 such that u factors through

k k (α) CK = {f ∈ C (K): f vanishes on ∂K for all α with |α| ≤ k}

Fix a smooth compactly-supported function ψ identically 1 on a neighborhood of 0, bounded between 0 and 1, and (necessarily) identically 0 outside some (larger) neighborhood of 0. For ε > 0 let

−1 ψε(x) = ψ(ε x)

n Since the support of u is just {0}, for all ε > 0 and for all f ∈ D(R ) the support of f − ψε · f does not include 0, so u(ψε · f) = u(f) Thus, for some constant C (depending on k and K, but not on f)

(α) −|j| (j) −1 (i−j) |ψεf|k = sup sup |(ψεf) (x)| ≤ C · sup sup sup ε ψ (ε x) f (x) x∈K |α|≤k |i|≤k x 0≤j≤i

For f vanishing to order k at 0, that is, f (α)(0) = 0 for all multi-indices α with |α| ≤ k, on a ﬁxed neighborhood of 0, by a Taylor-Maclaurin expansion, for some constant C

|f(x)| ≤ C · |x|k+1 and, generally, for αth derivatives with |α| ≤ k,

|f (α)(x)| ≤ C · |x|k+1−|α|

For some constant C

−|j| k+1−|i|+|j| k+1−|i| k+1−k |ψεf|k ≤ C · sup sup ε · ε ≤ C · ε ≤ C · ε = C · ε |i|≤k 0≤j≤i

Thus, for all ε > 0, for smooth f vanishing to order k at 0,

|u(f)| = |u(ψεf)| ≤ C · ε

Thus, u(f) = 0 for such f. That is, u is 0 on the intersection of the kernels of δ and its derivatives δ(α) for |α| ≤ k. Generally,

10 Paul Garrett: The Hilbert transform (February 14, 2017)

[8.2] Proposition: A continuous linear function λ ∈ V ∗ vanishing on the intersection of the kernels of a ﬁnite collection λ1, . . . , λn of continuous linear functionals on V is a linear combination of the λi.

Proof: The linear map n q : V −→ C by v −→ (λ1v, . . . , λnv) is continuous since each λi is continuous, and λ factors through q, as λ = L ◦ q for some linear functional L on Cn. We know all the linear functionals on Cn, namely, L is of the form

L(z1, . . . , zn) = c1z1 + ... + cnzn (for some constants ci)

Thus, λ(v) = (L ◦ q)(v) = L(λ1v, . . . , λnv) = c1λ1(v) + ... + cnλn(v)

expressing λ as a linear combination of the λi. ///

9. Appendix: the snake lemma

The snake lemma produces a long exact sequence from a short exact sequence of complexes, where in this context a complex is a chain of homomorphisms

f2 f1 f0 f−1 ... / M1 / M0 / M−1 / ...

[1] [2] where Imfi ⊂ ker fi1 for all indices. For a slightly specialized version of the snake lemma, consider a short exact sequences of very small complexes of the form 0 → A → A → 0, 0 → B → B → 0, and 0 → C → C → 0, with the complexes themselves now written vertically:

0 0 0

0 / A / B / C / 0

T T T 0 / A / B / C / 0

0 0 0

We assume that the maps T : A → A, T : B → B, and T : C → C make the squares in the latter diagram commute. In the following, one should consider the improbability of deﬁning some sort of map from any part of C to any part of A, given short exact 0 → A → B → C → 0:

[9.1] Claim: There is a long exact sequence

A B C 0 −→ ker T | −→ ker T | −→ ker T | −→ −→ −→ −→ 0 A B C TA TB TC

[1] There is also an assertion of naturality, which we ignore here. See [Weibel 1994] as general reference for these and related ideas.

[2] I ﬁrst saw this small but important example in [Casselman 1993].

11 Paul Garrett: The Hilbert transform (February 14, 2017)

Proof: The most serious issue, which is all we address, is defining the connecting homomorphism δ : ker T |C → A/T A. For example, the exactness of the left half and of the right half of the long sequence [3] are relatively elementary. Given c ∈ ker T |C , we want to determine a ∈ A well-defined module TA, to eventually fit into the long exact sequence. Let b → c for some b ∈ B. Since T c = 0 ∈ C and the squares commute, T b → 0 under the map B → C. Thus, by exactness of 0 → A → B → C → 0, there is a ∈ A mapping to T b. Declare a = δc. To make δ well-defined, note that the precise ambiguity in choice of a ∈ A is by TA. ///

Bibliography [Bedrosian 1962] E. Bedrosian, A product theorem for Hilbert transforms, Memoranudm RM-3439-PR, December 1962, Rand Corp., Santa Monica, CA. [Calderon-Zygmund 1952] A.P. Calderon, A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85-139. [Casselman 1993] W. Casselman, Extended automorphic forms on the upper half-plane, Math. Ann. 296 (1993), 755-762.

[Chaudhury 2012] K. N. Chaudhury, Lp-boundedness of the Hilbert transform, arXiv:0999.1426v9 [cs.IT] 2 Oct 2012. [Fefferman 1971] C. Fefferman, Characterization of bounded mean oscillation, Bull. AMS 77 (1971), 587-588. [Fefferman-Stein 1972] C. Fefferman, E. Stein, Hp spaces of several variables, Acta Math. 129 (1972), 137-193. [Garrett 2016a] P. Garrett, Vector-valued integrals, http://www.math.umn.edu/egarrett/m/fun/notes-2016-17/vector-valued-integrals.pdf [Garrett 2016b] P. Garrett, Holomorphic vector-valued functions, http://www.math.umn.edu/egarrett/m/fun/notes-2016-17/holomorphic-vector-valued.pdf [Gelfand-Silov 1964] I.M. Gelfand, G.E. Silov, Generalized Functions, I: Properties and Operators, Academic Press, NY, 1964. [Pichorides 1972] S. K. Pichorides, On the best values of the constants in the theorems of M. Riesz, Zygmund, and Kolmogorov, Studia Math. 44 (1972), 165-179.

[Plemelj 1908] J. Plemelj, Ein Eg¨anzungssatzzur Cauchyschen Integraldarstellung analytischer Funktionen, Randwerte betreﬀend, Monatsch. Math. Phys. 19 (1908), 205-10. [Plemelj 1908b] J. Plemelj, Riemannsche Funktionenscharen mit gegebener Monodromiegruppe, Monatsch. Math. Phys. 19 (1908), 211-245.

[Riesz 1928] M. Riesz, Sur les fonctions conjug´ees, Math. Z. (1928), 218-244. [Sokhotskii 1873] Y.W. Sokhotskii, On deﬁnite integrals and functions used in series expansion, (Russian?) St. Petersburg, 1873. [Wiki 2015] https://en.wikipedia.org/wiki/Sokhotski-Plemelj theorem, accessed 08 Oct 2015.

[Stein 1970] E. Stein, Singular integrals and Diﬀerentiability Properties of Functions, Princeton U. Press,

[3] The exactness of those halves also follows from the slightly-less elementary fact that right adjoints are left exact, and left adjoints are right exact.

12 Paul Garrett: The Hilbert transform (February 14, 2017)

1970. [Weibel 1994] C. Weibel, An introduction to homological algebra, Cambridge Univ. Press, 1994. [Wiki 2014] Wikipedia, Hilbert transform, https://en.wikipedia.org/wiki/Hilbert transform, 20 Dec 2014. [Woodward-Davies 1952] P.M. Woodward, I.L. Davies, Information theory and inverse probability in telecommunication, Proc. Inst. Elec. Engrs. Part III 99 (1952), 37-44. [Zygmund 1968] A. Zygmund, Trigonometric Series I, II, Cambridge, 1968.