The spectral measures can be used to extend the continuous functional calculus to bounded Borel functions.
3.
This measure is sometimes called the "'spectral measure associated to h " '.
4.
The limit of the empirical spectral measure for Wigner matrices was described by Eugene Wigner; see Wigner semicircle distribution.
5.
Same result holds for a discrete-time stationary process, with the spectral measure now defined on the unit circle.
6.
The limit of the empirical spectral measure of invariant matrix ensembles is described by a certain integral equation which arises from potential theory.
7.
The limit of the empirical spectral measure of Wishart matrices was found by Vladimir Marchenko and Leonid Pastur, see Marchenko Pastur distribution.
8.
Let " H " ac be the subspace consisting of vectors whose spectral measures are absolutely continuous with respect to the Lebesgue measure.
9.
The Riesz-Markov theorem then allows us to pass from integration on continuous functions to spectral measures, and this is the Borel functional calculus.
10.
Via its spectral measures, one can define a decomposition of the spectrum of any self adjoint operator, bounded or otherwise into absolutely continuous, pure point, and singular parts.
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