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-Stieltjestransform, the Fourier transform exists if and only if the function to be transformed is Lebesgue integrable over the whole real axis for $t\in ℝ$, or over the entire $ℂ$ domain when $\check{m}(t)$ is a complex function.

Definition 0.1.

Fourier-Stieltjes transform.

Given a positive definite, measurable function $f(x)$ on the interval$(-\mathrm{\infty },\mathrm{\infty })$ there exists a monotone increasing, real-valued boundedfunction $\alpha (t)$ such that:

$$f(x)={\int }_{ℝ}{e}^{itx}d(\alpha (t),$$(0.1)

for all $x\in ℝ$ except a small set. When $f(x)$ is defined as above and if $\alpha (t)$ is nondecreasing and bounded then the measurable function defined by the above integral is called the Fourier-Stieltjes transform of $\alpha (t)$, and it is continuous in addition to being positive definite.

FT Generalizations