orthonormal

SearchAdvancedHelp

(0.001 seconds)

1—10 of 15 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

… ►Let $p_n(\xi )$, $n=0,1,\mathrm{\dots }$, be the orthonormal set of polynomials defined by …

3: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►The corresponding eigenvectors ${𝐚}_1,\mathrm{\dots },{𝐚}_n$ can be chosen such that they form a complete orthonormal basis in ${𝐄}_n$. … ►

Orthonormal Expansions

►Assuming $\{{𝐚}_i\}$ is an orthonormal basis in ${𝐄}_n$, any vector $𝐮$ may be expanded as …

4: 33.14 Definitions and Basic Properties

… ►

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

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: 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

ⓘ

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: 18.2 General Orthogonal Polynomials

… ►(iii) orthonormal OP’s: $h_n=1$ (and usually, but not always, $k_n>0$); … ►

Monic and Orthonormal Forms

… ►In terms of the monic OP’s $p_n$ define the orthonormal OP’s $q_n$ by …Then, with the coefficients (18.2.11_4) associated with the monic OP’s $p_n$, the orthonormal recurrence relation for $q_n$ takes the form … ►The monic and orthonormal OP’s, and their determination via recursion, are more fully discussed in §§3.5(v) and 3.5(vi), where modified recursion coefficients are listed for the classical OP’s in their monic and orthonormal forms. …

7: 18.36 Miscellaneous Polynomials

… ►Exceptional type I $X_m$-EOP’s, form a complete orthonormal set with respect to a positive measure, but the lowest order polynomial in the set is of order $m$, or, said another way, the first $m$ polynomial orders, $0,1,\mathrm{\dots },m-1$ are missing. … ►and orthonormal with respect to the weight function …

8: 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 $L^2(-\tau ,\tau )$. …

9: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►A (finite or countably infinite, generalizing the definition of (1.2.40)) set $\{v_n\}$ is an orthonormal set if the $v_n$ are normalized and pairwise orthogonal. … ►For an orthonormal set $\{v_n\}$ in a Hilbert space $V$ Bessel’s inequality holds: … ►Such orthonormal sets are called complete. … ► The analogous orthonormality is … ►If an eigenvalue is of multiplicity greater than $1$ then an orthonormal basis of eigenfunctions can be given for the eigenspace. …

10: 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 … ►Instead of orthonormal polynomials $\{{\varphi }_n(z)\}$ Simon (2005a, b) uses monic polynomials ${\mathrm{\Phi }}_n(z)$. …Then the corresponding orthonormal polynomials are …