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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 f0g 9. Appendix: the snake lemma Formulaically, the Cauchy principal-value functional η is Z 1 f(x) Z f(x) ηf = principal-value functional of f = P:V: dx = lim dx −∞ x "!0+ jxj>" 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 1 f(x) Z 1 f(x) lim dx = −iπ f(0) + P:V: dx "!0+ −∞ x + i" −∞ 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 2 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 2 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 1 f(t) 1 Z f(t) (Hf)(x) = P:V: dt = lim dt π −∞ x − t π "!0+ jt−xj>" 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 1 f(x) Z f(x) ηf = principal-value functional of f = P:V: dx = lim dx −∞ x "!0+ jxj>" x We prove that η is a tempered distribution: [1.1] Claim: For f 2 S , for every h > 0, Z f(x) 0 jη(f)j ≤ dx + 2h sup jf (x)j jx|≥h x jx|≤h Thus, u is a tempered distribution: jη(f)j sup(1 + x2)jf(x)j + sup jf 0(x)j (implied constant independent of f) x x Proof: Certainly Z f(x) Z f(x) jη(f)j = dx + lim dx + jx|≥h x "!0 "<jx|≤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 "<jx|≤h x "<jx|≤h x "<jx|≤h "<jx|≤h because 1=x is odd and x ! f(0) is even. Thus, Z f(x) Z Z dx = f(ξx) dx ≤ jf(ξx)j dx "<jx|≤h x "<jx|≤h "<jx|≤h ≤ 2h · sup jf 0(x)j ≤ 2h · sup jf 0(x)j "<jx|≤h jx|≤h The latter is independent of ". With 0 < h ≤ 1, Z f(x) Z f(x) Z f(x) dx = dx + dx jx|≥h x jx|≥1 x h≤|x|≤1 x Z Z Z 1 2 1 ≤ jf(x)j dx + jf(x)j dx ≤ (1 + x )jf(x)j · 2 dx + 2h sup jf(x)j jx|≥1 h h≤|x|≤1 jx|≥1 1 + x x Z 2 1 2 ≤ sup(1 + x )jf(x)j · 2 dx + 2h sup jf(x)j sup(1 + x )jf(x)j x jx|≥1 1 + x x x 0 0 Of course, supjx|≤h jf (x)j ≤ supx jf (x)j, giving the corollary. === 2 Paul Garrett: The Hilbert transform (February 14, 2017) To make the degree of a positive-homogeneous function ' agree with the degree of the integration-against-' distribution u', due to change-of-measure, we find that Z Z −1 −1 (u' ◦ t)(f) = u'(f ◦ t ) = u'(x) f(t x) dx = u'(tx) f(x) d(tx) = t · u'◦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 "th 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) "!0+ jx|≥" x "!0+ jx|≥" −x Z f(x) Z f(x) = lim − dx = − lim dx = −η(f) "!0+ jx|≥" x "!0+ jx|≥" 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 "!0+ jx|≥" x t "!0+ jtx|≥" tx t "!0+ jx|≥"=t x 1 Z f(x) 1 = lim d(x) = η(f) t "!0+ jx|≥" x t That is, η is homogeneous of degree −1. === 2. Other descriptions of the principal-value functional [2.1] Claim: Let ' be any even function in L1(R) \ C1(R), with '(0) = 1. Then Z f(x) Z f(x) − f(0) · '(x) P:V: dx = dx (for f 2 Co( ) \ L1( ), for example) x x R R R R where f(x) − f(0) · '(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 '. f(x)−f(0)·'(x) Extending x by continuity at 0, Z f(x) − f(0) · '(x) Z f(x) − f(0) · '(x) dx = lim dx x "!0+ x R jx|≥" Z f(x) Z '(x) Z f(x) = lim dx + f(0) lim dx = P:V: dx + 0 "!0+ x "!0+ x x jx|≥" jx|≥" 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 1 f(x) Z 1 f(x) lim dx = −iπ f(0) + P:V: dx "!0+ −∞ x + i" −∞ 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 "!0+ x + i" "!0+ x + i" "!0+ x + i" R R R For 0 < " < 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 + i" x=i (1 + x2)(x + i") (i + i)(i + i") 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 1 f(x) 1 dx = P:V: dx (for f 2 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 "!0+ x "!0+ x R jx|≥" jx|≥" 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.
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