The Titchmarsh theorem states that if is square integrable over the real -axis, then any one of the following implies the other two:

1. The Fourier transform 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

(Bracewell 1999, Problem 8, p. 273).

See also

Fourier Transform, Hilbert Transform

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References

Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 1999.

Referenced on Wolfram|Alpha

Titchmarsh Theorem

Cite this as:

Weisstein, Eric W. "Titchmarsh Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TitchmarshTheorem.html

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