orthonormal
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9 matching pages
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: 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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8: 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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9: 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.