# orthonormal

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## 10 matching pages

##### 1: 28.30 Expansions in Series of Eigenfunctions
Let $\widehat{\lambda}_{m}$, $m=0,1,2,\dots$, be the set of characteristic values (28.29.16) and (28.29.17), arranged in their natural order (see (28.29.18)), and let $w_{m}(x)$, $m=0,1,2,\dots$, be the eigenfunctions, that is, an orthonormal set of $2\pi$-periodic solutions; thus …
##### 2: 32.15 Orthogonal Polynomials
Let $p_{n}(\xi)$, $n=0,1,\dots$, 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 $c_{n}=a_{n-1}$ ($n\geq 1$). …
##### 4: 33.14 Definitions and Basic Properties
33.14.15 $\int_{0}^{\infty}\phi_{m,\ell}(r)\phi_{n,\ell}(r)\,\mathrm{d}r=\delta_{m,n}.$
Note that the functions $\phi_{n,\ell}$, $n=\ell,\ell+1,\ldots$, do not form a complete orthonormal system. …
##### 5: 31.15 Stieltjes Polynomials
31.15.12 $\rho(z)=\left(\prod_{j=1}^{N-1}\prod_{k=1}^{N}|z_{j}-a_{k}|^{\gamma_{k}-1}% \right)\left(\prod_{j
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 $\phi_{n}$, $n=0,1,2,\dots$ 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 $\ell^{2}$ by choosing a complete orthonormal set $\left\{v_{n}\right\}_{n=0}^{\infty}$ in $V$ . Then an isomorphism is given byAssume that $\left\{\phi_{n}\right\}_{n=0}^{\infty}$ 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 $\{\phi_{n}(z)\}$, $n=0,1,\dots$, where $\phi_{n}(z)$ is of proper degree $n$, is orthonormal on the unit circle with respect to the weight function $w(z)$ ($\geq 0$) if …
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. …
Just below (33.14.9), the constraint described in the text “$\ell<(-\epsilon)^{-1/2}$ when $\epsilon<0$,” was removed. In Equation (33.14.13), the constraint $\epsilon_{1},\epsilon_{2}>0$ was added. In the line immediately below (33.14.13), it was clarified that $s\left(\epsilon,\ell;r\right)$ 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 $\phi_{n,\ell}(r)$ to Laguerre polynomials. A sentence was added immediately below (33.14.15) indicating that the functions $\phi_{n,\ell}$, $n=\ell,\ell+1,\ldots$, do not form a complete orthonormal system.