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Hilbert himself was surprised by the unexpected application of this theory, noting that "I developed my theory of infinitely many variables from purely mathematical interests, and even called it 'spectral analysis' without any presentiment that it would later find application to the actual spectrum of physics."
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 spectral theorem for self-adjoint operators in particular that underlies much of the existing Hilbert space theory was generalized to C*-algebras. [27] These techniques are now basic in abstract harmonic analysis and representation theory.
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 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.
In functional analysis, every C *-algebra is isomorphic to a subalgebra of the C *-algebra () of bounded linear operators on some Hilbert space. This article describes the spectral theory of closed normal subalgebras of B ( H ) {\displaystyle {\mathcal {B}}(H)} .
In functional analysis, compact operators are linear operators on Banach spaces that map bounded sets to relatively compact sets. In the case of a Hilbert space H, the compact operators are the closure of the finite rank operators in the uniform operator topology. In general, operators on infinite-dimensional spaces feature properties that do ...
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 .