Titchmarsh Theorem (original) (raw)

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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 F(t)=F_omega[f(omega)](t) is 0 for t<0 .

2. Replacing omega by z=x+iy , the function f(z) is analytic in the complex plane for y>0 and approaches f(x) almost everywhere as y->0 . Furthermore, int_(-infty)^infty|f(x+iy)|^2dx<k for some number and y>0 (i.e., the integral is bounded).

3. The real and imaginary parts of F(z) 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.

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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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