About the Project

orthonormal

AdvancedHelp

(0.000 seconds)

10 matching pages

1: 28.30 Expansions in Series of Eigenfunctions
Let λ ^ m , m = 0 , 1 , 2 , , be the set of characteristic values (28.29.16) and (28.29.17), arranged in their natural order (see (28.29.18)), and let w m ( x ) , m = 0 , 1 , 2 , , be the eigenfunctions, that is, an orthonormal set of 2 π -periodic solutions; thus …
2: 32.15 Orthogonal Polynomials
Let p n ( ξ ) , n = 0 , 1 , , be the orthonormal set of polynomials defined by …
3: 18.2 General Orthogonal Polynomials
then two special normalizations are: (i) orthonormal OP’s: h n = 1 , k n > 0 ; (ii) monic OP’s: k n = 1 . … If the OP’s are orthonormal, then c n = a n 1 ( n 1 ). …
4: 33.14 Definitions and Basic Properties
33.14.15 0 ϕ m , ( r ) ϕ n , ( r ) d r = δ m , n .
Note that the functions ϕ n , , n = , + 1 , , do not form a complete orthonormal system. …
5: 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 ) . …
6: 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 ) . …
7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
A Hilbert space V is separable if there is an (at most countably infinite) orthonormal set { v n } in V such that for every v V where c n is given by (1.18.3). Such orthonormal sets are called complete. By (1.18.4)Every infinite dimensional separable Hilbert space V can be made isomorphic to 2 by choosing a complete orthonormal set { v n } n = 0 in V . Then an isomorphism is given byAssume that { ϕ n } n = 0 is an orthonormal basis of L 2 ( X ) . The formulas in §1.18(i) are then: The analogous orthonormality is
8: 18.33 Polynomials Orthogonal on the Unit Circle
A system of polynomials { ϕ n ( z ) } , n = 0 , 1 , , where ϕ n ( z ) is of proper degree n , is orthonormal on the unit circle with respect to the weight function w ( z ) ( 0 ) if …
9: 3.5 Quadrature
The corresponding orthonormal polynomials q n ( x ) = p n ( x ) / h n satisfy the recurrence relation … The monic and orthonormal recursion relations of this section are both closely related to the Lanczos recursion relation in §3.2(vi). … The monic version p n ( x ) and orthonormal version q n ( x ) of a classical orthogonal polynomial are obtained by dividing the orthogonal polynomial by k n respectively h n , with k n and h n as in Table 18.3.1. …
Table 3.5.17_5: Recurrence coefficients in (3.5.30) and (3.5.30_5) for monic versions p n ( x ) and orthonormal versions q n ( x ) of the classical orthogonal polynomials.
p n ( x ) q n ( x ) α n β n h 0
10: Errata
  • Subsection 33.14(iv)

    Just below (33.14.9), the constraint described in the text “ < ( ϵ ) 1 / 2 when ϵ < 0 ,” was removed. In Equation (33.14.13), the constraint ϵ 1 , ϵ 2 > 0 was added. In the line immediately below (33.14.13), it was clarified that s ( ϵ , ; r ) is exp ( r / n ) times a polynomial in r / n , instead of simply a polynomial in r . In Equation (33.14.14), a second equality was added which relates ϕ n , ( r ) to Laguerre polynomials. A sentence was added immediately below (33.14.15) indicating that the functions ϕ n , , n = , + 1 , , do not form a complete orthonormal system.