# orthonormal

(0.000 seconds)

## 10 matching pages

##### 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

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

►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
$$

►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.
…
►
…

##### 10: Errata

…
►
Subsection 33.14(iv)
…

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.