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Titchmarsh Theorem

If $f(\omega)$ is Square Integrable over the Real $\omega$ axis, then any one of the following implies the other two:

1. The Fourier Transform of $f(\omega)$ is 0 for $t < 0$.

2. Replacing $\omega$ by $z$, the function $f(z)$ is analytic in the Complex Plane $z$ for $y > 0$ and approaches $f(x)$ almost everywhere as $y\to 0$. Furthermore, $\int_{-\infty}^\infty \vert f(x+iy)\vert^2\,dx < k$ for some number $k$ and $y > 0$ (i.e., the integral is bounded).

3. The Real and Imaginary Parts of $f(z)$ are Hilbert Transforms of each other.

© 1996-9 Eric W. Weisstein