orthonormal
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1: 28.30 Expansions in Series of Eigenfunctions
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►Let , , be the set of characteristic values (28.29.16) and (28.29.17), arranged in their natural order (see (28.29.18)), and let , , be the eigenfunctions, that is, an orthonormal set of -periodic solutions; thus
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2: 32.15 Orthogonal Polynomials
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►Let , , be the orthonormal set of polynomials defined by
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3: 18.2 General Orthogonal Polynomials
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►then two special normalizations are: (i) orthonormal OP’s: , ; (ii) monic OP’s: .
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►If the OP’s are orthonormal, then ().
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4: 33.14 Definitions and Basic Properties
5: 31.15 Stieltjes Polynomials
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31.15.12
►The normalized system of products (31.15.8) forms an orthonormal basis in the Hilbert space .
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6: 30.15 Signal Analysis
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►The sequence , forms an orthonormal basis in the space of -bandlimited functions, and, after normalization, an orthonormal basis in .
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7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
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►A Hilbert space
is separable if there is an (at most countably
infinite) orthonormal set
in
such that for every
…where
is given by (1.18.3).
Such orthonormal sets are called complete.
By (1.18.4)
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►Every infinite dimensional separable Hilbert space
can be made isomorphic
to
by choosing a complete
orthonormal set
in
. Then an isomorphism is given by
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►Assume that
is an orthonormal basis of
.
The formulas in §1.18(i) are then:
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The analogous orthonormality is
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8: 18.33 Polynomials Orthogonal on the Unit Circle
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►A system of polynomials , , where is of proper degree , is orthonormal on the unit circle with respect
to the weight function
() if
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9: 3.5 Quadrature
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►The corresponding orthonormal polynomials satisfy the recurrence relation
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►The monic and orthonormal recursion relations of this section are both closely related to the Lanczos recursion relation in §3.2(vi).
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►The monic version and orthonormal version of a classical orthogonal polynomial are obtained by dividing the orthogonal polynomial by respectively , with and as in Table 18.3.1.
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10: Errata
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Subsection 33.14(iv)
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Just below (33.14.9), the constraint described in the text “ when ,” was removed. In Equation (33.14.13), the constraint was added. In the line immediately below (33.14.13), it was clarified that is times a polynomial in , instead of simply a polynomial in . In Equation (33.14.14), a second equality was added which relates to Laguerre polynomials. A sentence was added immediately below (33.14.15) indicating that the functions , , do not form a complete orthonormal system.