<p>(February 14, 2017) </p><p>The Hilbert transform </p><p>Paul Garrett&nbsp;[email protected] http:/<a href="/goto?url=http://www.math.umn.edu/" target="_blank">/www.math.umn.edu/</a>egarrett/ <br>[This document is http:/<a href="/goto?url=http://www.math.umn.edu/" target="_blank">/www.math.umn.edu/</a>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 </p><p>Formulaically, the Cauchy principal-value functional η is </p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p>∞</p><p></p><ul style="display: flex;"><li style="flex:1">f(x) </li><li style="flex:1">f(x) </li></ul><p></p><ul style="display: flex;"><li style="flex:1">ηf = principal-value&nbsp;functional of f = P.V. </li><li style="flex:1">dx = lim </li></ul><p></p><p>dx x</p><p>ε→0<sup style="top: -0.1661em;">+ </sup></p><p>x</p><p>−∞ </p><p>|x|&gt;ε </p><p>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: </p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">∞</li><li style="flex:1">∞</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">f(x) </li><li style="flex:1">f(x) </li></ul><p>lim </p><p></p><ul style="display: flex;"><li style="flex:1">dx = −iπ f(0) + P.V. </li><li style="flex:1">dx </li></ul><p></p><p>ε→0<sup style="top: -0.1661em;">+ </sup></p><p></p><ul style="display: flex;"><li style="flex:1">x + iε </li><li style="flex:1">x</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">−∞ </li><li style="flex:1">−∞ </li></ul><p></p><p>See also [Gelfand-Silov 1964]. Other plausible identities are best certified using uniqueness, such as </p><p>Z</p><p>f(x) − f(−x) </p><p>12</p><p></p><ul style="display: flex;"><li style="flex:1">ηf = </li><li style="flex:1">dx </li></ul><p></p><p>(for f ∈ C<sup style="top: -0.3013em;">1</sup>(R) ∩ L<sup style="top: -0.3013em;">1</sup>(R)) </p><p>x</p><p>R</p><p>f(x)−f(−x) </p><p></p><ul style="display: flex;"><li style="flex:1">using the canonical continuous extension of </li><li style="flex:1">at 0. Also, </li></ul><p></p><p>x</p><p>Z</p><p>2</p><p>f(x) − f(0) · e<sup style="top: -0.3013em;">−x </sup></p><p></p><ul style="display: flex;"><li style="flex:1">ηf = </li><li style="flex:1">dx </li></ul><p></p><p>(for f ∈ C<sup style="top: -0.3012em;">o</sup>(R) ∩ L<sup style="top: -0.3012em;">1</sup>(R)) </p><p>x</p><p>R</p><p>The Hilbert transform of a function f on R is awkwardly described as a principal-value integral </p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p>∞</p><p>1<br>(Hf)(x) =&nbsp;P.V. </p><p>π</p><p></p><ul style="display: flex;"><li style="flex:1">f(t) </li><li style="flex:1">1</li><li style="flex:1">f(t) </li></ul><p></p><p>dt = </p><p>lim </p><p>dt x − t π <sup style="top: 0.0117em;">ε</sup>→0<sup style="top: -0.1544em;">+ </sup>x − t </p><p>|t−x|&gt;ε </p><p>−∞ </p><p>with the leading constant 1/π understandable with sufficient hindsight:&nbsp;we will see that this adjustment makes H extend to a unitary operator on L<sup style="top: -0.3013em;">2</sup>(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. </p><p>1</p><p>Paul Garrett: The Hilbert transform (February 14, 2017) </p><p>1. The principal-value functional </p><p>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).&nbsp;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. </p><p>Again, let </p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p>∞</p><p></p><ul style="display: flex;"><li style="flex:1">f(x) </li><li style="flex:1">f(x) </li></ul><p></p><ul style="display: flex;"><li style="flex:1">ηf = principal-value&nbsp;functional of f = P.V. </li><li style="flex:1">dx = lim </li></ul><p></p><p>dx x</p><p>ε→0<sup style="top: -0.1661em;">+ </sup></p><p>x</p><p>−∞ </p><p>|x|&gt;ε </p><p>We prove that η is a tempered distribution: [1.1] Claim: For f ∈ S , for every h &gt; 0, </p><p>|η(f)| ≤ </p><p>Z<br>ꢀ</p><p>ꢀꢀ<br>ꢀꢀꢀ</p><p>f(x) dx + 2h sup |f<sup style="top: -0.3428em;">0</sup>(x)| </p><p>x</p><p></p><ul style="display: flex;"><li style="flex:1">|x|≤h </li><li style="flex:1">|x|≥h </li></ul><p></p><p>Thus, u is a tempered distribution: </p><ul style="display: flex;"><li style="flex:1">|η(f)| ꢀ sup(1 + x<sup style="top: -0.3428em;">2</sup>)|f(x)| + sup |f<sup style="top: -0.3428em;">0</sup>(x)| </li><li style="flex:1">(implied constant independent of f) </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">x</li><li style="flex:1">x</li></ul><p></p><p>Proof: Certainly </p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p>ꢀ</p><p>ꢀꢀ<br>ꢀꢀꢀ<br>ꢀꢀꢀ<br>ꢀꢀꢀ</p><p></p><ul style="display: flex;"><li style="flex:1">f(x) </li><li style="flex:1">f(x) </li></ul><p></p><ul style="display: flex;"><li style="flex:1">|η(f)| = </li><li style="flex:1">dx + lim </li></ul><p></p><p>dx x</p><p>ε→0<sup style="top: -0.166em;">+ </sup></p><p>x</p><p>|x|≥h </p><p>ε&lt;|x|≤h </p><p>By the Mean Value Theorem, f(x) = f(0) + x f<sup style="top: -0.3012em;">0</sup>(ξ<sub style="top: 0.1245em;">x</sub>) for some ξ<sub style="top: 0.1245em;">x </sub>between 0 and x. Thus, </p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">f(x) </li><li style="flex:1">f(0) </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">dx = </li><li style="flex:1">dx + </li></ul><p></p><p>f(ξx) dx = 0&nbsp;+ </p><p>f(ξx) dx </p><ul style="display: flex;"><li style="flex:1">x</li><li style="flex:1">x</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">ε&lt;|x|≤h </li><li style="flex:1">ε&lt;|x|≤h </li><li style="flex:1">ε&lt;|x|≤h </li><li style="flex:1">ε&lt;|x|≤h </li></ul><p></p><p>because 1/x is odd and x → f(0) is even. Thus, </p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p>ꢀ</p><p>ꢀꢀ<br>ꢀꢀꢀ<br>ꢀꢀꢀ<br>ꢀꢀꢀ</p><p>f(x) </p><p>dx </p><p>=</p><p>f(ξx) dx </p><p>≤</p><p>|f(ξx)| dx x</p><p></p><ul style="display: flex;"><li style="flex:1">ε&lt;|x|≤h </li><li style="flex:1">ε&lt;|x|≤h </li><li style="flex:1">ε&lt;|x|≤h </li></ul><p></p><p>≤ 2h · sup |f<sup style="top: -0.3428em;">0</sup>(x)| ≤ 2h · sup |f<sup style="top: -0.3428em;">0</sup>(x)| </p><p>ε&lt;|x|≤h </p><p>|x|≤h </p><p>The latter is independent of ε. With 0 &lt; h ≤ 1, </p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p>ꢀ</p><p>ꢀꢀ<br>ꢀꢀꢀ<br>ꢀꢀꢀ<br>ꢀꢀꢀ<br>ꢀꢀꢀ<br>ꢀ</p><p></p><ul style="display: flex;"><li style="flex:1">f(x) </li><li style="flex:1">f(x) </li><li style="flex:1">f(x) </li></ul><p></p><p>ꢀ</p><p></p><ul style="display: flex;"><li style="flex:1">dx = </li><li style="flex:1">dx + </li><li style="flex:1">dx </li></ul><p></p><p>ꢀ</p><p></p><ul style="display: flex;"><li style="flex:1">x</li><li style="flex:1">x</li><li style="flex:1">x</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">|x|≥h </li><li style="flex:1">|x|≥1 </li><li style="flex:1">h≤|x|≤1 </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">1</li></ul><p></p><p>≤</p><p>|f(x)| dx + </p><p>|f(x)| dx ≤ </p><p></p><ul style="display: flex;"><li style="flex:1">(1 + x<sup style="top: -0.3428em;">2</sup>)|f(x)| · </li><li style="flex:1">dx + 2h sup |f(x)| </li></ul><p></p><p>h</p><p>1 + x<sup style="top: -0.2398em;">2 </sup></p><p>x</p><p></p><ul style="display: flex;"><li style="flex:1">|x|≥1 </li><li style="flex:1">h≤|x|≤1 </li><li style="flex:1">|x|≥1 </li></ul><p></p><p>Z</p><p>1</p><ul style="display: flex;"><li style="flex:1">≤ sup(1 + x<sup style="top: -0.3427em;">2</sup>)|f(x)| · </li><li style="flex:1">dx + 2h sup |f(x)| ꢀ sup(1 + x<sup style="top: -0.3427em;">2</sup>)|f(x)| </li></ul><p>1 + x<sup style="top: -0.2398em;">2 </sup></p><p></p><ul style="display: flex;"><li style="flex:1">x</li><li style="flex:1">x</li><li style="flex:1">x</li></ul><p></p><p>|x|≥1 </p><p>Of course, sup<sub style="top: 0.2029em;">|x|≤h </sub>|f<sup style="top: -0.3013em;">0</sup>(x)| ≤ sup<sub style="top: 0.2029em;">x </sub>|f<sup style="top: -0.3013em;">0</sup>(x)|, giving the corollary. </p><p>/// </p><p>2</p><p>Paul Garrett: The Hilbert transform (February 14, 2017) </p><p>To make the degree of a positive-homogeneous function ϕ agree with the degree of the integration-against-ϕ distribution u<sub style="top: 0.1245em;">ϕ</sub>, due to change-of-measure, we find that </p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p>(u<sub style="top: 0.1245em;">ϕ </sub>◦ t)(f) = u<sub style="top: 0.1245em;">ϕ</sub>(f ◦ t<sup style="top: -0.3427em;">−1</sup>) = </p><p>u<sub style="top: 0.1245em;">ϕ</sub>(x) f(t<sup style="top: -0.3427em;">−1</sup>x) dx = </p><p>u<sub style="top: 0.1245em;">ϕ</sub>(tx) f(x) d(tx) = t · u<sub style="top: 0.1245em;">ϕ◦t</sub>(f) </p><p></p><ul style="display: flex;"><li style="flex:1">R</li><li style="flex:1">R</li></ul><p></p><p>Thus, for agreement of the notion of homogeneity for distributions and (integrate-against) functions, the dilation action u → u ◦ t on distributions should be </p><p>1<br>(u ◦ t)(f) =&nbsp;u(f ◦ t<sup style="top: -0.3428em;">−1 </sup></p><ul style="display: flex;"><li style="flex:1">)</li><li style="flex:1">(for t &gt; 0, test function f, and distribution u) </li></ul><p></p><p>t</p><p>Parity is as expected:&nbsp;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. </p><p>[1.2] Claim: The principal-value distribution η is positive-homogeneous of degree −1, and is odd. Proof: The ε<sup style="top: -0.3013em;">th </sup>integral in the limit definition of η is itself odd, by changing variables: </p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">f(−x) </li><li style="flex:1">f(x) </li></ul><p>η(f ◦ (−1)) =&nbsp;lim dx = lim d(−x) </p><p>ε→0<sup style="top: -0.166em;">+ </sup></p><p>x</p><p>ε→0<sup style="top: -0.166em;">+ </sup></p><p>−x </p><p></p><ul style="display: flex;"><li style="flex:1">|x|≥ε </li><li style="flex:1">|x|≥ε </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">f(x) </li><li style="flex:1">f(x) </li></ul><p>=</p><ul style="display: flex;"><li style="flex:1">lim − </li><li style="flex:1">dx = − lim </li></ul><p></p><p>dx = −η(f) </p><p>ε→0<sup style="top: -0.166em;">+ </sup></p><p>x</p><p>ε→0<sup style="top: -0.166em;">+ </sup></p><p>x</p><p></p><ul style="display: flex;"><li style="flex:1">|x|≥ε </li><li style="flex:1">|x|≥ε </li></ul><p></p><p>as claimed. For the degree of homogeneity, for t &gt; 0 and test function f, </p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">f(<sup style="top: -0.3268em;">x</sup><sub style="top: 0.2863em;">t </sub>) </li><li style="flex:1">1</li><li style="flex:1">1</li><li style="flex:1">1</li><li style="flex:1">1</li><li style="flex:1">f(x) </li><li style="flex:1">1</li><li style="flex:1">f(x) </li></ul><p>(η ◦t)(f) =&nbsp;η(f ◦ ) = lim </p><p>dx = </p><p>Z</p><p></p><ul style="display: flex;"><li style="flex:1">lim </li><li style="flex:1">d(tx) = </li><li style="flex:1">lim </li><li style="flex:1">d(x) </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">t</li><li style="flex:1">t</li><li style="flex:1">t <sup style="top: 0.0117em;">ε</sup>→0<sup style="top: -0.1544em;">+ </sup></li><li style="flex:1">x</li><li style="flex:1">t <sup style="top: 0.0117em;">ε</sup>→0<sup style="top: -0.1544em;">+ </sup></li><li style="flex:1">tx </li><li style="flex:1">t <sup style="top: 0.0117em;">ε</sup>→0<sup style="top: -0.1544em;">+ </sup></li><li style="flex:1">x</li></ul><p></p><p>|x|≥ε </p><p></p><ul style="display: flex;"><li style="flex:1">|tx|≥ε </li><li style="flex:1">|x|≥ε/t </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">f(x) </li><li style="flex:1">1</li></ul><p></p><ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">lim </li><li style="flex:1">d(x) =&nbsp;η(f) </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">t <sup style="top: 0.0117em;">ε</sup>→0<sup style="top: -0.1544em;">+ </sup></li><li style="flex:1">x</li><li style="flex:1">t</li></ul><p></p><p>|x|≥ε </p><p>That is, η is homogeneous of degree −1. </p><p>/// </p><p>2. Other descriptions of the principal-value functional </p><p>[2.1] Claim: Let ϕ be any even function in L<sup style="top: -0.3013em;">1</sup>(R) ∩ C<sup style="top: -0.3013em;">1</sup>(R), with ϕ(0) = 1.&nbsp;Then </p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">f(x) </li><li style="flex:1">f(x) − f(0) · ϕ(x) </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">P.V. </li><li style="flex:1">dx = </li><li style="flex:1">dx </li></ul><p></p><p>(for f ∈ C<sup style="top: -0.3013em;">o</sup>(R) ∩ L<sup style="top: -0.3013em;">1</sup>(R), for example) </p><p></p><ul style="display: flex;"><li style="flex:1">x</li><li style="flex:1">x</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">R</li><li style="flex:1">R</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul><p></p><p>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. </p><p>f(x)−f(0)·ϕ(x) </p><p></p><ul style="display: flex;"><li style="flex:1">Extending </li><li style="flex:1">by continuity at 0, </li></ul><p></p><p>x</p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">f(x) − f(0) · ϕ(x) </li><li style="flex:1">f(x) − f(0) · ϕ(x) </li></ul><p>dx = lim </p><p>dx x</p><p>ε→0<sup style="top: -0.1661em;">+ </sup></p><p>x</p><p>R</p><p>|x|≥ε </p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">f(x) </li><li style="flex:1">ϕ(x) </li><li style="flex:1">f(x) </li></ul><p></p><ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">lim </li><li style="flex:1">dx + f(0) lim </li></ul><p></p><p>dx = P.V. </p><p>dx + 0 </p><p>ε→0<sup style="top: -0.1661em;">+ </sup></p><p>x</p><p>ε→0<sup style="top: -0.1661em;">+ </sup></p><p></p><ul style="display: flex;"><li style="flex:1">x</li><li style="flex:1">x</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">|x|≥ε </li><li style="flex:1">|x|≥ε </li></ul><p></p><p>R</p><p>as claimed. </p><p>/// </p><p>3</p><p>Paul Garrett: The Hilbert transform (February 14, 2017) </p><p>The leading term in the following might be unexpected: </p><p>[2.2] Claim: (Sokhotski-Plemelj) </p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">∞</li><li style="flex:1">∞</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">f(x) </li><li style="flex:1">f(x) </li></ul><p>lim </p><p></p><ul style="display: flex;"><li style="flex:1">dx = −iπ f(0) + P.V. </li><li style="flex:1">dx </li></ul><p></p><p>ε→0<sup style="top: -0.166em;">+ </sup></p><p></p><ul style="display: flex;"><li style="flex:1">x + iε </li><li style="flex:1">x</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">−∞ </li><li style="flex:1">−∞ </li></ul><p></p><p>Proof: Aiming to apply the previous claim, </p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p>f(0) 1+x<sup style="top: -0.166em;">2 </sup></p><p>f(x) − </p><ul style="display: flex;"><li style="flex:1">f(x) </li><li style="flex:1">1/(1 + x<sup style="top: -0.3012em;">2</sup>) </li></ul><p></p><ul style="display: flex;"><li style="flex:1">lim </li><li style="flex:1">dx = lim </li><li style="flex:1">dx + f(0) lim </li></ul><p></p><p>dx </p><p>ε→0<sup style="top: -0.1661em;">+ </sup></p><p>x + iε </p><p>ε→0<sup style="top: -0.1661em;">+ </sup></p><p>x + iε </p><p>ε→0<sup style="top: -0.1661em;">+ </sup></p><p>x + iε </p><p></p><ul style="display: flex;"><li style="flex:1">R</li><li style="flex:1">R</li><li style="flex:1">R</li></ul><p></p><p>For 0 &lt; ε &lt; 1, the right-most integral is evaluated by residues, moving the contour through the upper half-plane: </p><p>Z</p><p></p><ul style="display: flex;"><li style="flex:1">1/(1 + x<sup style="top: -0.3012em;">2</sup>) </li><li style="flex:1">1</li><li style="flex:1">1</li></ul><p></p><p>πidx = 2πi Res<sub style="top: 0.1245em;">x=i </sub></p><p>= 2πi </p><p>−→ </p><p>= −iπ x + iε </p><p></p><ul style="display: flex;"><li style="flex:1">(1 + x<sup style="top: -0.2399em;">2</sup>)(x + iε) </li><li style="flex:1">(i + i)(i + iε) </li></ul><p></p><p>R</p><p>as claimed. [2.3] Claim: With f continuously differentiable at 0, thereby extending </p><p>/// </p><p>f(x)−f(−x) </p><p>by continuity at 0, </p><p>x</p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p>∞</p><p></p><ul style="display: flex;"><li style="flex:1">f(x) − f(−x) </li><li style="flex:1">f(x) </li></ul><p></p><p>12</p><p></p><ul style="display: flex;"><li style="flex:1">dx = P.V. </li><li style="flex:1">dx </li></ul><p></p><p>(for f ∈ C<sup style="top: -0.3012em;">1</sup>(R) ∩ L<sup style="top: -0.3012em;">1</sup>(R)) </p><p></p><ul style="display: flex;"><li style="flex:1">x</li><li style="flex:1">x</li></ul><p></p><p>R</p><p>−∞ </p><p>f(x)−f(−x) </p><p>Proof: For test function f, since functional, is continuous as 0, using the odd-ness of the principal-value </p><p>x</p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">f(x) − f(−x) </li><li style="flex:1">f(x) − f(−x) </li></ul><p>f(x) </p><p>12<br>12</p><p>dx = </p><p></p><ul style="display: flex;"><li style="flex:1">lim </li><li style="flex:1">dx = lim </li></ul><p></p><p>dx x</p><p>R</p><p>ε→0<sup style="top: -0.1661em;">+ </sup></p><p>x</p><p>ε→0<sup style="top: -0.1661em;">+ </sup></p><p>x</p><p></p><ul style="display: flex;"><li style="flex:1">|x|≥ε </li><li style="flex:1">|x|≥ε </li></ul><p></p><p>as claimed. </p><p>/// </p><p>3. Uniqueness of odd/even homogeneous distributions </p><p>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. </p><p>d</p><p>[3.1] Claim: For a positive-homogeneous distribution u of degree s, x<sub style="top: 0.2863em;">dx </sub>u is again positive-homogeneous of degree s, and with the same parity. </p><p>Proof: For test function f, </p><p>d</p><p>1</p><p>d</p><p>1</p><p>d</p><p>1</p><p>d</p><p></p><ul style="display: flex;"><li style="flex:1">((x u) ◦ t)(f) =&nbsp;(x u)(f ◦ t<sup style="top: -0.3427em;">−1</sup>) = </li><li style="flex:1">(</li><li style="flex:1">u)(x · f ◦ t<sup style="top: -0.3427em;">−1</sup>) = − u( (x · (f ◦ t<sup style="top: -0.3427em;">−1</sup>))) </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">dx </li><li style="flex:1">t</li><li style="flex:1">dx </li><li style="flex:1">t dx </li><li style="flex:1">t</li><li style="flex:1">dx </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">1</li><li style="flex:1">1</li></ul><p></p><ul style="display: flex;"><li style="flex:1">= − u(f ◦ t<sup style="top: -0.3427em;">−1 </sup>+ x · t<sup style="top: -0.3427em;">−1 </sup>· (f<sup style="top: -0.3427em;">0 </sup>◦ t<sup style="top: -0.3427em;">−1</sup>)) = − u(f ◦ t<sup style="top: -0.3427em;">−1 </sup>+ (x · f<sup style="top: -0.3427em;">0</sup>) ◦ t<sup style="top: -0.3427em;">−1</sup>)) = − u((f + xf<sup style="top: -0.3427em;">0</sup>) ◦ t<sup style="top: -0.3427em;">−1 </sup></li><li style="flex:1">)</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">t</li><li style="flex:1">t</li><li style="flex:1">t</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">ꢁ</li><li style="flex:1">ꢂ</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">d</li><li style="flex:1">d</li><li style="flex:1">d</li></ul><p></p><p>= −(u ◦ t)(f + xf<sup style="top: -0.3428em;">0</sup>) = −(u ◦ t)( (xf)) = t<sup style="top: -0.3428em;">s </sup>· −&nbsp;u( (xf)) =&nbsp;t<sup style="top: -0.3428em;">s </sup>· (x u)(f) </p><p></p><ul style="display: flex;"><li style="flex:1">dx </li><li style="flex:1">dx </li><li style="flex:1">dx </li></ul><p></p><p>4</p><p>Paul Garrett: The Hilbert transform (February 14, 2017) </p><p>as asserted.&nbsp;Preservation of parity is similar:&nbsp;writing f<sup style="top: -0.3013em;">−</sup>(x) = f(−x) for functions f, and u<sup style="top: -0.3013em;">−</sup>(f) = u(f<sup style="top: -0.3013em;">−</sup>) for distributions u, </p><p></p><ul style="display: flex;"><li style="flex:1">d</li><li style="flex:1">d</li><li style="flex:1">d</li><li style="flex:1">d</li><li style="flex:1">d</li></ul><p></p><p>(x u)<sup style="top: -0.3427em;">−</sup>(f) = (x u)(f<sup style="top: -0.3427em;">−</sup>) = (&nbsp;u)(x · f<sup style="top: -0.3427em;">−</sup>) = −u( (x · f<sup style="top: -0.3427em;">−</sup>)) = u( ((x · f)<sup style="top: -0.3427em;">−</sup>)) </p><p></p><ul style="display: flex;"><li style="flex:1">dx </li><li style="flex:1">dx </li><li style="flex:1">dx </li><li style="flex:1">dx </li><li style="flex:1">dx </li></ul><p></p><p>ꢁ</p><p>ꢂ<sub style="top: 0.1721em;">− </sub></p><p></p><ul style="display: flex;"><li style="flex:1">d</li><li style="flex:1">d</li><li style="flex:1">d</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">= u(− </li><li style="flex:1">(x · f) )&nbsp;= −u<sup style="top: -0.3427em;">−</sup>( (x · f)) = ((x )u<sup style="top: -0.3427em;">−</sup>)(f) </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">dx </li><li style="flex:1">dx </li><li style="flex:1">dx </li></ul><p></p><p>as claimed. </p><p>/// </p><p>d</p><p>The Euler operator x<sub style="top: 0.2862em;">dx </sub>is the infinitesimal form of dilation f(x) → f(tx). This is manifest in the following: </p>