Fourier-Stieltjes transforms and measured groupoid transforms are useful generalizations of the (much simpler) Fourier transform, as concisely shown in the following table (see also Fourier transforms
)
- for the purpose of direct comparison with the latter transform. Unlike the more general Fourier-Stieltjes transform, the Fourier transform exists if and only if the function to be transformed is Lebesgue integrable over the whole real axis for
, or over the entire
domain when
is a complex function.
Given a positive definite, measurable function
on the interval
there exists a monotone increasing, real-valued bounded
function
such that:
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(0.1) |
for all
except a small set. When
is defined as above and if
is nondecreasing and bounded then the measurable function defined by the above integral is called the Fourier-Stieltjes transform of
, and it is continuous in addition to being positive definite.
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Conditions* | Explanation | Description |
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from ![]() ![]() |
From
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Notice on the next line the overline | bar (
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Fourier-Stieltjes transform |
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locally compact groupoid [1]; | ||||
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a left Haar measure
on ![]() |
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as above | Inverse Fourier-Stieltjes |
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transform | ([2], [3]). | |||
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When
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This is the usual |
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only when
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Inverse Fourier transform | |||
Lebesgue integrable on | ||||
the entire real axis |
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