# orthonormal

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## 9 matching pages

##### 1: 28.30 Expansions in Series of Eigenfunctions

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►Let ${\widehat{\lambda}}_{m}$, $m=0,1,2,\mathrm{\dots}$, be the set of characteristic values (28.29.16) and (28.29.17), arranged in their natural order (see (28.29.18)), and let ${w}_{m}(x)$, $m=0,1,2,\mathrm{\dots}$, be the

*eigenfunctions*, that is, an orthonormal set of $2\pi $-periodic solutions; thus …##### 2: 32.15 Orthogonal Polynomials

##### 3: 18.2 General Orthogonal Polynomials

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►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 ${c}_{n}={a}_{n-1}$ ($n\ge 1$). …##### 4: 33.14 Definitions and Basic Properties

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33.14.15
$${\int}_{0}^{\mathrm{\infty}}{\varphi}_{m,\mathrm{\ell}}(r){\varphi}_{n,\mathrm{\ell}}(r)dr={\delta}_{m,n}.$$

►Note that the functions ${\varphi}_{n,\mathrm{\ell}}$, $n=\mathrm{\ell},\mathrm{\ell}+1,\mathrm{\dots}$, do not form a complete orthonormal system.
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##### 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 ${L}_{\rho}^{2}(Q)$.
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##### 6: 30.15 Signal Analysis

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►The sequence ${\varphi}_{n}$, $n=0,1,2,\mathrm{\dots}$ forms an orthonormal basis in the space of $\sigma $-bandlimited functions, and, after normalization, an orthonormal basis in ${L}^{2}(-\tau ,\tau )$.
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##### 7: 18.33 Polynomials Orthogonal on the Unit Circle

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►A system of polynomials $\{{\varphi}_{n}(z)\}$, $n=0,1,\mathrm{\dots}$, where ${\varphi}_{n}(z)$ is of proper degree $n$, is

*orthonormal on the unit circle with respect to the weight function*$w(z)$ ($\ge 0$) if …##### 8: 3.5 Quadrature

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►The corresponding orthonormal polynomials ${q}_{n}(x)={p}_{n}(x)/\sqrt{{h}_{n}}$ 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 ${p}_{n}(x)$ and orthonormal version ${q}_{n}(x)$ of a classical orthogonal polynomial are obtained by dividing the orthogonal polynomial by ${k}_{n}$ respectively $\sqrt{{h}_{n}}$, with ${k}_{n}$ and ${h}_{n}$ 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 ${\u03f5}_{1},{\u03f5}_{2}>0$ was added. In the line immediately below (33.14.13), it was clarified that $s(\u03f5,\mathrm{\ell};r)$ is $\mathrm{exp}\left(-r/n\right)$ 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 ${\varphi}_{n,\mathrm{\ell}}(r)$ to Laguerre polynomials. A sentence was added immediately below (33.14.15) indicating that the functions ${\varphi}_{n,\mathrm{\ell}}$, $n=\mathrm{\ell},\mathrm{\ell}+1,\mathrm{\dots}$, do not form a complete orthonormal system.