Since and are isomorphisms, we know that,,,, is a complete orthogonal system in.
Moreover, the corresponding associated family of eigenfunctions ({psi_{beta}}) is a complete orthogonal set in (W^{2,2}).
Hence { e ( ± q n 1 − q x ; q 2 ), n ∈ Z } is a complete orthogonal set in L 2 ( R ˜ q ).
Since { e 2 k π i t | k ∈ Z } is a complete orthogonal system of L 2 ( I ), every h ∈ L 2 ( I ) can be expressed by the Fourier series expansion h ( t ) = ∑ k = − ∞ ∞ h k ⋅ e 2 k π i t, where h k = ∫ 0 1 h ( s ) e 2 k π i s d s, k ∈ Z.
The second extension of WSK sampling theorem is the theorem of Kramer, [11] which states that if I is a finite closed interval, K ( ⋅, t ) : I × C → C is a function continuous in t such that K ( ⋅, t ) ∈ L 2 ( I ) for all t ∈ C. Let { t k } k ∈ Z be a sequence of real numbers such that { K ( ⋅, t k ) } k ∈ Z is a complete orthogonal set in L 2 ( I ).
Denote (X=H^{2}cap W) and its finite-dimensional subspaces X^{m}=operatorname{span}bigl{ Phi^{1},Phi^{2}, ldots,Phi^{m}bigr},quad m=1,2,ldots, where ({Phi^{m}}_{m=1}^{infty}) is a complete normal orthogonal basis of X.
It is well known that for the self-adjoint compact operators the eigenvalues are real, and the corresponding eigenfunctions form a complete orthogonal basis on (L^{2}).
17 and 18), ξ i is a set of normal random variables, n KL is the number of terms of the truncated discretization, f i (t) are a complete set of deterministic orthogonal functions and λ i are the eigenvalues of the covariance function C(t 1,t 2 ): C(t_{1},t_{2} ) = e^{{ - left| {t_{1} - t_{2} } right|/b}} (20 where b is the correlation length, which must be expressed in the same units as t.
It happens yet that experimental material is limited and it does not allow using a complete (orthogonal) SPSB design.
A bounded set Ω⊂Rd is called a spectral set if the space L2 admits a complete orthogonal system of exponential functions.
