table of generalized Fourier and measured groupoid transforms
0.1 Generalized Fourier transforms
Fourier-Stieltjes transforms and measured groupoid transforms are useful generalizations of the (much simpler) Fourier transform, as concisely shown in the following table- with the same format as C. Woo’s Feature on Fourier transforms (http://planetmath.org/TableOfFourierTransforms) - 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.
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.
|Title||table of generalized Fourier and measured groupoid transforms|
|Date of creation||2013-03-22 18:10:27|
|Last modified on||2013-03-22 18:10:27|
|Last modified by||bci1 (20947)|
|Defines||Fourier-Stieltjes and measured groupoid transforms|