Hilbert Transform

$$\begin{eqnarray*} g(y) &=& {1\over \pi} \int_{-\infty}^\infty{f(x)\,dx\over x-y... ... f(x)&=& {1\over \pi} \int_{-\infty}^\infty {g(y)\,dy\over y-x}. \end{eqnarray*}$$

See also Titchmarsh Theorem

References

Bracewell, R. The Fourier Transform and Its Applications. New York: McGraw-Hill, pp. 267-272, 1965.