0.1 Generalized Fourier transforms
Fourier–Stieltjes transforms and measured groupoid transforms are useful generalizations
of the Fourier transform, as summarized in the following table. See also Fourier transforms for
comparison with the ordinary Fourier transform.
Unlike the more general Fourier–Stieltjes transform, the ordinary Fourier transform requires
appropriate integrability conditions on the function being transformed.
Definition 0.1. Fourier–Stieltjes transform.
Given a positive-definite measurable function f(x) on (−∞,∞), there exists a monotone
increasing, real-valued, bounded function α(t) such that
for all x ∈ ℝ except possibly on a small exceptional set. When α(t) is nondecreasing and bounded,
the measurable function defined by the integral above is called the Fourier–Stieltjes transform of α;
it is continuous in addition to being positive definite.
Fourier transforms and generalizations
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| Input | Transform | Conditions | Explanation | Description |
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| f(t) | ℱ{f(t)} = f(x) =
(2π)−1 ∫
e−itx dx | Conditions depend
on the transform
convention and on
the function f. | Ordinary Fourier
transform | f(x) |
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| e−t𝜃(t) | ℱ{f(t)}(x) =
(2π)−1 ∫
𝜃(t)eit2x dx | From −∞ to +∞ | Numerical
example | Calculated with
Mathematica |
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| c | ( )−1c | | | |
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| f(t) | ∫
f(x)t(x)dx | f(t) ∈ L1(Gl),
with Gl a locally
compact groupoid
[1]; the integral is
defined using a left
Haar measure on
Gl. | Fourier–Stieltjes
transform | f(x) ∈ C0(Gl) |
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| m(x) | m(t) = ∫
eitx dm(x) | As above | Inverse
Fourier–Stieltjes
transform | m(t) ∈ L1(Gl);
see [2, 3]. |
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| m(x) | m(t) = ∫
eitx dm(x) | When G
l = ℝ,
and when m(x) is
Lebesgue integrable
on the entire real
axis | Usual inverse
Fourier transform | m(t) ∈ ℝ |
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The overline in t(x) denotes complex conjugation.
The numerical example above was calculated using this Mathematica demonstration.
References
[1] A. Ramsay and M. E. Walter, “Fourier–Stieltjes algebras of locally compact
groupoids,” Journal of Functional Analysis, 148: 314–367 (1997).
[2] A. L. T. Paterson, “The Fourier algebra for locally compact groupoids,” preprint
(2001).
[3] A. L. T. Paterson, “The Fourier–Stieltjes and Fourier algebras for locally compact
groupoids,” (2003). Free PDF file download
