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1: 28.30 Expansions in Series of Eigenfunctions

… ►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

… ►Let $p_n(\xi )$, $n=0,1,\mathrm{\dots }$, 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 $c_n=a_{n-1}$ ($n\ge 1$). …

4: 33.14 Definitions and Basic Properties

… ►

33.14.15 $${\int }_0^{\mathrm{\infty }}{\varphi }_{m,\mathrm{\ell }}(r){\varphi }_{n,\mathrm{\ell }}(r)dr={\delta }_{m,n}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential, $\int$: integral, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and ${\varphi }_{n,\mathrm{\ell }}(r)$: function

Proof sketch:

Derivable from (33.14.14) by combining it with Dunkl (2003, Theorem 2.2).

Referenced by:

§13.10(vi), §13.23(v), §33.14(iv), Erratum (V1.0.24) for Equation (33.14.15), Erratum (V1.0.24) for Subsection 33.14(iv)

Permalink:

http://dlmf.nist.gov/33.14.E15

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.24):

This equation, originally written as a definite integral ${\int }_0^{\mathrm{\infty }}{\varphi }_{n,\mathrm{\ell }}^2(r)dr=1$, was rewritten more generally as an orthonormality relation.

See also:

Annotations for §33.14(iv), §33.14 and Ch.33

►Note that the functions ${\varphi }_{n,\mathrm{\ell }}$, $n=\mathrm{\ell },\mathrm{\ell }+1,\mathrm{\dots }$, do not form a complete orthonormal system. …

5: 31.15 Stieltjes Polynomials

… ►

31.15.12

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Defines:

$\rho (z)$: weight function (locally)

Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.15.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(iii), §31.15 and Ch.31

►The normalized system of products (31.15.8) forms an orthonormal basis in the Hilbert space $L_{\rho }^2(Q)$. …

6: 30.15 Signal Analysis

… ►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 $L^2(-\tau ,\tau )$. …

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 $v\in V$ …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 ${\mathrm{\ell }}^2$ by choosing a complete orthonormal set ${\left\{v_n\right\}}_{n=0}^{\mathrm{\infty }}$ in $V$ . Then an isomorphism is given by … ►Assume that ${\left\{{\varphi }_n\right\}}_{n=0}^{\mathrm{\infty }}$ is an orthonormal basis of $L^2\left(X\right)$ . 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 $\{{\varphi }_n(z)\}$, $n=0,1,\mathrm{\dots }$, 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 …

9: 3.5 Quadrature

… ►The corresponding orthonormal polynomials $q_n(x)=p_n(x)/\sqrt{h_n}$ 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 $\sqrt{h_n}$, 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.

► ►►

$p_n(x)$	$q_n(x)$	${\alpha }_n$	${\beta }_n$	$h_0$
…

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…

10: Errata

… ►

Just below (33.14.9), the constraint described in the text “ when ,” 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 $s(ϵ,\mathrm{\ell };r)$ is $\exp \left(-r/n\right)$ 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.

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