If is Square Integrable over the Real axis, then any one of the following implies the other two:

- 1. The Fourier Transform of is 0 for .
- 2. Replacing by , the function is analytic in the Complex Plane for and approaches almost everywhere as . Furthermore, for some number and (i.e., the integral is bounded).
- 3. The Real and Imaginary Parts of are Hilbert Transforms of each other.

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1999-05-26