Let be a positive definite, measurable function on the interval . Then there exists a monotone increasing, real-valued bounded function such that

for "almost all" . If is nondecreasing and bounded and is defined as above, then is called the Fourier-Stieltjes transform of , and is both continuous and positive definite.

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

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Weisstein, Eric W. "Fourier-Stieltjes Transform." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Fourier-StieltjesTransform.html