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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:
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.
In mathematics, particularly in functional analysis, a projection-valued measure (or spectral measure) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. [1]
The Hilbert spectrum (sometimes referred to as the Hilbert amplitude spectrum), named after David Hilbert, is a statistical tool that can help in distinguishing among a mixture of moving signals. The spectrum itself is decomposed into its component sources using independent component analysis .
When studying a closed unbounded operator A: H → H on a Hilbert space H, if there exists () such that (;) is a compact operator, we say that A has compact resolvent. The spectrum () of such A is a discrete subset of .
The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics. The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional ...
The spectral theorem is the beginning of the vast research area of functional analysis called operator theory; see also spectral measure. There is also an analogous spectral theorem for bounded normal operators on Hilbert spaces. The only difference in the conclusion is that now may be complex-valued.
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 ...