inverse hilbert transform
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Brief Word note by Phil, dated 4.3.13 (PhL), written for his spectral theory book. It treats the Hilbert transform as a convolution in frequency space that becomes multiplication by i sgn(t) in t space, and uses the Fourier transform of sgn(t) to argue that applying the transform twice gives minus the identity, so the inverse is minus the transform. He says it is a very early note, probably absorbed into Appendix C(h). The extracted equations are partly garbled.
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Inverting the Hilbert Transform PhL 4.3.13
Very early notes on this subject, I think all has been absorbed into Appendix C (h).
Xh(ω) ≡ – (1/π) dω'
xh(t) = i sgn(t) x(t)
This is a diagonalized convolution equation in t space. Recall our inverse theorem
A(ω) = (1/2π) !Syntax Error, Idω' B(ω-ω')C(ω') a(t) = b(t) c(t) (3.7)
Xh(ω) = (1/2π) !Syntax Error, Idω' SGN(ω-ω')i X(ω') xh(t) = i sgn(t) x(t) (3.7)
Now what is the FT of sgn(t)?
SGN(ω) = !Syntax Error, Idt sgn(t) e-iωt = -2i!Syntax Error, Idt sin(ωt) = 2i/ω I think
Then we get
Xh(Xh(ω) = - ω => Xh-1(ω) = – Xh(ω)