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1: 30.15 Signal Analysis
The sequence ϕ n , n = 0 , 1 , 2 , forms an orthonormal basis in the space of σ -bandlimited functions, and, after normalization, an orthonormal basis in L 2 ( τ , τ ) . …
2: 31.15 Stieltjes Polynomials
31.15.12 ρ ( z ) = ( j = 1 N 1 k = 1 N | z j a k | γ k 1 ) ( j < k N 1 ( z k z j ) ) .
The normalized system of products (31.15.8) forms an orthonormal basis in the Hilbert space L ρ 2 ( Q ) . …
3: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
Assume that { ϕ n } n = 0 is an orthonormal basis of L 2 ( X ) . … The eigenfunctions form a complete orthogonal basis in L 2 ( X ) , and we can take the basis as orthonormal: …
4: 18.39 Applications in the Physical Sciences
For either sign of Z , and s chosen such that n + l + 1 + ( 2 Z / s ) > 0 , n = 0 , 1 , 2 , , truncation of the basis to N terms, with x i N [ 1 , 1 ] , the discrete eigenvectors are the orthonormal L 2 functions …
5: 1.3 Determinants, Linear Operators, and Spectral Expansions
If 𝐷 n [ a j , k ] tends to a limit L as n , then we say that the infinite determinant 𝐷 [ a j , k ] converges and 𝐷 [ a j , k ] = L . … The corresponding eigenvectors 𝐚 1 , , 𝐚 n can be chosen such that they form a complete orthonormal basis in 𝐄 n . …
Orthonormal Expansions
Assuming { 𝐚 i } is an orthonormal basis in 𝐄 n , any vector 𝐮 may be expanded as …