linear functional

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1: 1.16 Distributions

… ►A mapping $\mathrm{\Lambda }:𝒟(I)\to ℂ$ is a linear functional if …$\mathrm{\Lambda }:𝒟(I)\to ℂ$ is called a distribution, or generalized function, if it is a continuous linear functional on $𝒟(I)$, that is, it is a linear functional and for every ${\varphi }_n\to \varphi$ in $𝒟(I)$, … ►A tempered distribution is a continuous linear functional $\mathrm{\Lambda }$ on $𝒯$. … ►A distribution in ${ℝ}^n$ is a continuous linear functional on ${𝒟}_n$. … ►Tempered distributions are continuous linear functionals on this space of test functions. …

2: 15.19 Methods of Computation

… ►For $z\in ℝ$ it is always possible to apply one of the linear transformations in §15.8(i) in such a way that the hypergeometric function is expressed in terms of hypergeometric functions with an argument in the interval $[0,\frac{1}{2}]$. … ►This is because the linear transformations map the pair $\{{\mathrm{e}}^{\pi \mathrm{i}/3},{\mathrm{e}}^{-\pi \mathrm{i}/3}\}$ onto itself. … ► …

3: 35.11 Tables

… ►Each table expresses the zonal polynomials as linear combinations of monomial symmetric functions.

4: 21.8 Abelian Functions

… ►For every Abelian function, there is a positive integer $n$, such that the Abelian function can be expressed as a ratio of linear combinations of products with $n$ factors of Riemann theta functions with characteristics that share a common period lattice. …

5: 2.6 Distributional Methods

… ►, a continuous linear functional) on the space $𝒯$ of rapidly decreasing functions on $ℝ$. …

6: 37.2 General Orthogonal Polynomials of Two Variables

… ►In the other direction, as an analogue of Favard’s theorem (see §18.2(viii) for the one-variable case), any polynomial system that satisfies the three-term relations (37.2.7), together with the conditions (37.2.10) and (37.2.8) of the coefficient matrices, must be orthonormal with respect to a positive definite linear functional. …

7: Bibliography G

… ►

The Computation of Special Functions by Linear Difference Equations. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 213–243.

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External Links:

ISBN 90-5699-521-9, MathReview, ZentralBlatt

Referenced by:

§3.6(iii), §3.6(vii).

Encodings:

BibTeX

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8: 37.20 Mathematical Applications

… ►In the latter method, the approximating functions are taken as a linear combinations of OPs and their coefficients are determined by the Galerkin method. …

9: 7.21 Physical Applications

… ►Fried and Conte (1961) mentions the role of $w\left(z\right)$ in the theory of linearized waves or oscillations in a hot plasma; $w\left(z\right)$ is called the plasma dispersion function or Faddeeva (or Faddeyeva) function; see Faddeeva and Terent’ev (1954). …

10: 15.8 Transformations of Variable

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§15.8(i) Linear Transformations

… ►

15.8.12 $$𝐅(a,b;a+b-m;z)={(1-z)}^{-m}𝐅(\tilde{a},\tilde{b};\tilde{a}+\tilde{b}+m;z),$$ $\tilde{a}=a-m,\tilde{b}=b-m$.

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Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $m$: integer

Permalink:

http://dlmf.nist.gov/15.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(ii), §15.8 and Ch.15

… ►A quadratic transformation relates two hypergeometric functions, with the variable in one a quadratic function of the variable in the other, possibly combined with a fractional linear transformation. … ►The transformation formulas between two hypergeometric functions in Group 2, or two hypergeometric functions in Group 3, are the linear transformations (15.8.1). …