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  2. Spectral theory - Wikipedia

    en.wikipedia.org/wiki/Spectral_theory

    The spectrum of T is the set of all complex numbers ζ such that R ζ fails to exist or is unbounded. Often the spectrum of T is denoted by σ(T). The function R ζ for all ζ in ρ(T) (that is, wherever R ζ exists as a bounded operator) is called the resolvent of T. The spectrum of T is therefore the complement of the resolvent set of T in ...

  3. Spectrum (functional analysis) - Wikipedia

    en.wikipedia.org/wiki/Spectrum_(functional_analysis)

    In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix.

  4. Decomposition of spectrum (functional analysis) - Wikipedia

    en.wikipedia.org/wiki/Decomposition_of_spectrum...

    The spectrum of T restricted to H ac is called the absolutely continuous spectrum of T, σ ac (T). The spectrum of T restricted to H sc is called its singular spectrum, σ sc (T). The set of eigenvalues of T is called the pure point spectrum of T, σ pp (T). The closure of the eigenvalues is the spectrum of T restricted to H pp.

  5. Spectrum of a matrix - Wikipedia

    en.wikipedia.org/wiki/Spectrum_of_a_matrix

    In mathematics, the spectrum of a matrix is the set of its eigenvalues. [ 1 ] [ 2 ] [ 3 ] More generally, if T : V → V {\displaystyle T\colon V\to V} is a linear operator on any finite-dimensional vector space , its spectrum is the set of scalars λ {\displaystyle \lambda } such that T − λ I {\displaystyle T-\lambda I} is not invertible .

  6. Spectrum (topology) - Wikipedia

    en.wikipedia.org/wiki/Spectrum_(topology)

    In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem .

  7. Essential spectrum - Wikipedia

    en.wikipedia.org/wiki/Essential_spectrum

    In mathematics, the essential spectrum of a bounded operator (or, more generally, of a densely defined closed linear operator) is a certain subset of its spectrum, defined by a condition of the type that says, roughly speaking, "fails badly to be invertible".

  8. Welch's method - Wikipedia

    en.wikipedia.org/wiki/Welch's_method

    Welch's method is an improvement on the standard periodogram spectrum estimating method and on Bartlett's method, in that it reduces noise in the estimated power spectra in exchange for reducing the frequency resolution. Due to the noise caused by imperfect and finite data, the noise reduction from Welch's method is often desired.

  9. Discrete spectrum (mathematics) - Wikipedia

    en.wikipedia.org/.../Discrete_spectrum_(Mathematics)

    A point in the spectrum of a closed linear operator: in the Banach space with domain is said to belong to discrete spectrum of if the following two conditions are satisfied: [1] λ {\displaystyle \lambda } is an isolated point in σ ( A ) {\displaystyle \sigma (A)} ;

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