TOPICS
Titchmarsh Theorem
If
issquare integrable over thereal
-axis, then any one of the following implies the other two:
1. TheFourier transform](/image.pl?url=https%3a%2f%2fmathworld.wolfram.com%2fimages%2fequations%2fTitchmarshTheorem%2fInline3.svg&f=jpg&w=240)
is 0 for
.
2. Replacing
by
, the function
is analytic in thecomplex plane
for
and approaches
almost everywhere as
. Furthermore,
for some number
and
(i.e., the integral is bounded).
3. Thereal andimaginary parts of
areHilbert transforms of each other
(Bracewell 1999, Problem 8, p. 273).
See also
Fourier Transform,HilbertTransformExplore with Wolfram|Alpha
More things to try:
References
Bracewell, R.The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 1999.Referenced on Wolfram|Alpha
Titchmarsh TheoremCite this as:
Weisstein, Eric W. "Titchmarsh Theorem."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/TitchmarshTheorem.html