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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 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.
Spectral analysis or spectrum analysis is analysis in terms of a spectrum ... a theory that extends eigenvalues and eigenvectors to linear operators on Hilbert ...
Linear Operators is a three-volume textbook on the theory of linear operators, written by Nelson Dunford and Jacob T. Schwartz. The three volumes are (I) General Theory; (II) Spectral Theory, Self Adjoint Operators in Hilbert Space; and (III) Spectral Operators. The first volume was published in 1958, the second in 1963, and the third in 1971.
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, 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.
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)} .