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Fourier-Stieltjes Transform

Let $f(x)$ be a positive definite, measurable function on the Interval $(-\infty, \infty)$. Then there exists a monotone increasing, real-valued bounded function $\alpha(t)$ such that

f(x)=\int_{-\infty}^\infty e^{itx}\,d\alpha(t)

for ``Almost All'' $x$. If $\alpha(t)$ is nondecreasing and bounded and $f(x)$ is defined as above, then $f(x)$ is called the Fourier-Stieltjes transform of $\alpha(t)$, and is both continuous and positive definite.

See also Fourier Transform, Laplace Transform


Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 618, 1980.

© 1996-9 Eric W. Weisstein