measure with jumps

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21: 3.2 Linear Algebra

… ►The sensitivity of the solution vector $𝐱$ in (3.2.1) to small perturbations in the matrix $𝐀$ and the vector $𝐛$ is measured by the condition number … ►If $𝐀$ is nondefective and $\lambda$ is a simple zero of $p_n(\lambda )$, then the sensitivity of $\lambda$ to small perturbations in the matrix $𝐀$ is measured by the condition number …

22: 18.28 Askey–Wilson Class

… ►If, in addition to (18.28.11) or (18.28.12), we have $a^{-1}b\le q$, then the measure in (18.28.10) is the unique orthogonality measure. … ►For continuous $q^{-1}$-Hermite polynomials the orthogonality measure is not unique. …

23: 1.16 Distributions

… ►More generally, for $\alpha :[a,b]\to [-\mathrm{\infty },\mathrm{\infty }]$ a nondecreasing function the corresponding Lebesgue–Stieltjes measure ${\mu }_{\alpha }$ (see §1.4(v)) can be considered as a distribution: … ►If the measure ${\mu }_{\alpha }$ is absolutely continuous with density $w$ (see §1.4(v)) then $𝐷\alpha ={\mathrm{\Lambda }}_w$. … ►Since ${\delta }_{x_0}$ is the Lebesgue–Stieltjes measure ${\mu }_{\alpha }$ corresponding to $\alpha (x)=H\left(x-x_0\right)$ (see §1.4(v)), formula (1.16.16) is a special case of (1.16.3_5), (1.16.9_5) for that choice of $\alpha$. … ►See Hildebrandt (1938) and Chihara (1978, Chapter II) for Stieltjes measures which are used in §18.39(iii); see also Shohat and Tamarkin (1970, Chapter II). …

24: 35.5 Bessel Functions of Matrix Argument

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35.5.8 $${\int }_{𝐎(m)}\mathrm{etr}\left(𝐒𝐇\right)\mathrm{d}𝐇=\frac{A_{-1/2}\left(-\frac{1}{4}𝐒{𝐒}^{\mathrm{T}}\right)}{A_{-1/2}\left(𝟎\right)},$$ $𝐒$ arbitrary.

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Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, $𝐒$: real symmetric $m\times m$ matrix, $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $\mathrm{d}𝐇$: normalized Haar measure on $𝐎(m)$ , $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.10, §35.9

Permalink:

http://dlmf.nist.gov/35.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(ii), §35.5 and Ch.35

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25: Bibliography G

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Stable computation of high order Gauss quadrature rules using discretization for measures in radiation transfer. J. Quant. Spectrosc. Radiat. Transfer 68 (2), pp. 213–223.

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Referenced by:

§18.39(iii).

Encodings:

BibTeX

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26: 18.3 Definitions

… ►However, in general they are not orthogonal with respect to a positive measure, but a finite system has such an orthogonality. …

27: 18.38 Mathematical Applications

… ►Exceptional OP’s (EOP’s) are those which are ‘missing’ a finite number of lower order polynomials, but yet form complete sets with respect to suitable measures. …

28: 18.27 $q$-Hahn Class

… ►The measure is not uniquely determined: … ►The measure is not uniquely determined: … ►For discrete $q$-Hermite II polynomials the measure is not uniquely determined. …

29: Bibliography C

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Extremal measures for a system of orthogonal polynomials. Constr. Approx. 9, pp. 111–119.

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External Links:

ISSN 0176-4276, Document, MathReview (Mizan Rahman), ZentralBlatt

Referenced by:

§18.28(iv).

Encodings:

BibTeX

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30: Bibliography M

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The supports of measures associated with orthogonal polynomials and the spectra of the related selfadjoint operators. Rocky Mountain J. Math. 21 (1), pp. 501–527.

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External Links:

ISSN 0035-7596, Document, Link, MathReview (T. S. Chihara)

Referenced by:

§18.2(xi).

Encodings:

BibTeX

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