orthonormal expansions

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1: 1.3 Determinants, Linear Operators, and Spectral Expansions

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Orthonormal Expansions

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2: Errata

… ►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. …

3: 31.15 Stieltjes Polynomials

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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 $L_{\rho }^2(Q)$. For further details and for the expansions of analytic functions in this basis see Volkmer (1999).

4: Bibliography K

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Hypergeometric expansions of Heun polynomials. SIAM J. Math. Anal. 22 (5), pp. 1450–1459.

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External Links:

ISSN 0036-1410, Document, MathReview (Peter A. McCoy), ZentralBlatt

Referenced by:

§31.16(ii).

Encodings:

BibTeX

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Addendum: “Hypergeometric expansions of Heun polynomials”. SIAM J. Math. Anal. 22 (6), pp. 1803.

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External Links:

ISSN 0036-1410, Document, MathReview (Peter A. McCoy), ZentralBlatt

Referenced by:

§31.16(ii).

Encodings:

BibTeX

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Series expansions for the third incomplete elliptic integral via partial fraction decompositions. J. Comput. Appl. Math. 207 (2), pp. 331–337.

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External Links:

ISSN 0377-0427, Document, MathReview (José Luis López), ZentralBlatt

Referenced by:

§19.12, §19.5.

Encodings:

BibTeX

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Orthonormal polynomials with generalized Freud-type weights. J. Approx. Theory 121 (1), pp. 13–53.

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External Links:

ISSN 0021-9045, Document, Link, MathReview (Walter Van Assche)

Referenced by:

§18.32.

Encodings:

BibTeX

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Asymptotic expansions of certain $q$-series and a formula of Ramanujan for specific values of the Riemann zeta function. Acta Arith. 107 (3), pp. 269–298.

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External Links:

ISSN 0065-1036, Document, Link, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§17.2(i).

Encodings:

BibTeX

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5: 28.30 Expansions in Series of Eigenfunctions

§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 …

6: 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. …

7: 18.39 Applications in the Physical Sciences

… ►These eigenfunctions are the orthonormal eigenfunctions of the time-independent Schrödinger equation … ►With the normalization factor ${\left(c{h}_n\right)}^{-1/2}$ the ${\psi }_n$ are orthonormal in $L^2(ℝ,dx)$. … ►The orthonormal stationary states and corresponding eigenvalues are then of the form …The finite system of functions ${\psi }_n$ is orthonormal in $L^2(ℝ,dx)$, see (18.34.7_3). … ►with an infinite set of orthonormal $L^2$ eigenfunctions …