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Eigenfunctions. In general, an eigenvector of a linear operator D defined on some vector space is a nonzero vector in the domain of D that, when D acts upon it, is simply scaled by some scalar value called an eigenvalue. In the special case where D is defined on a function space, the eigenvectors are referred to as eigenfunctions. That is, a ...
Eigenvalues and eigenvectors. In linear algebra, an eigenvector (/ ˈaɪɡən -/ EYE-gən-) or characteristic vector is a vector that has its direction unchanged by a given linear transformation. More precisely, an eigenvector, , of a linear transformation, , is scaled by a constant factor, , when the linear transformation is applied to it: .
Hilbert–Schmidt theorem. In mathematical analysis, the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert spaces. In the theory of partial differential equations, it is very useful in solving elliptic boundary value problems.
If the function κ is L 1 μ (X), where κ(x)=K(x,x), for all x in X, then there is an orthonormal set {e i} i of L 2 μ (X) consisting of eigenfunctions of T K such that corresponding sequence of eigenvalues {λ i} i is nonnegative. The eigenfunctions corresponding to non-zero eigenvalues are continuous on X and K has the representation
the first eigenvalue of the Laplace–Beltrami operator on the n-dimensional real projective space (with normalization given by the covering map from the unit-radius sphere) is 2n + 2, and the corresponding eigenfunctions are the restrictions of the homogeneous quadratic polynomials on R n + 1 to the unit sphere (and then to the real projective ...
Mercer's theorem states that the spectral decomposition of the integral operator of yields a series representation of in terms of the eigenvalues and eigenfunctions of . This then implies that K {\displaystyle K} is a reproducing kernel so that the corresponding RKHS can be defined in terms of these eigenvalues and eigenfunctions.
Hilbert–Schmidt operator. In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator that acts on a Hilbert space and has finite Hilbert–Schmidt norm. where is an orthonormal basis. [1][2] The index set need not be countable.
In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation, Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed ...