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The name spectral theory was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid , in an infinite-dimensional setting.
Template: Halmos Introduction to Hilbert Space and the Theory of Spectral Multiplicity 2017/doc
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Template: Halmos Introduction to Hilbert Space and the Theory of Spectral Multiplicity 2017
Hilbert spectral analysis is a signal analysis method applying the Hilbert transform to compute the instantaneous frequency of signals according to = (). After performing the Hilbert transform on each signal, we can express the data in the following form:
In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory.
The spectral theorem has many variants. A particularly powerful version is as follows: Theorem. For any Abelian von Neumann algebra A on a separable Hilbert space H, there is a standard Borel space X and a measure μ on X such that it is unitarily equivalent as an operator algebra to L ∞ μ (X) acting on a direct integral of Hilbert spaces
Throughout, is a fixed Hilbert space. A projection-valued measure on a measurable space (,), where is a σ-algebra of subsets of , is a mapping: such that for all , is a self-adjoint projection on (that is, () is a bounded linear operator (): that satisfies () = and () = ()) such that = (where is the identity operator of ) and for every ,, the function defined by (), is a complex measure on ...