test function space

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1: 1.16 Distributions

… ►The linear space of all test functions with the above definition of convergence is called a test function space. … … ►The space of all distributions will be denoted by

\mathcal{D}^{*}(I)

. … ►The space

\mathcal{T}(\mathbb{R})

of test functions for tempered distributions consists of all infinitely-differentiable functions such that the function and all its derivatives are

O\left({\left|x\right|}^{-N}\right)

\left|x\right|\to\infty

for all

N

. … ►Tempered distributions are continuous linear functionals on this space of test functions. …

$x, y$	real variables.
…
$L^{2}\left(X,\,\mathrm{d}\alpha\right)$	the space of all Lebesgue–Stieltjes measurable functions on $X$ which are square integrable with respect to $\,\mathrm{d}\alpha$ .
$\phi$	a testing function.
$\left\langle\Lambda,\phi\right\rangle$	action of distribution $\Lambda$ on test function $\phi$ .
…
$\mathbf{E}_{n}$	the space of all $n$ -dimensional vectors.
…

3: 2.6 Distributional Methods

… ►

2.6.11

\left\langle f,\phi\right\rangle=\int_{0}^{\infty}f(t)\phi(t)\,\mathrm{d}t,

\phi\in\mathcal{T}

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\in$ : element of, $\int$ : integral, $f(t)$ : locally integrable function and $\mathcal{T}$ : space of decreasing functions
Referenced by:: §2.6(ii)
Permalink:: http://dlmf.nist.gov/2.6.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §2.6(ii), §2.6 and Ch.2

… ►

2.6.12

\left\langle t^{-\alpha},\phi\right\rangle=\int_{0}^{\infty}t^{-\alpha}\phi(t)% \,\mathrm{d}t,

\phi\in\mathcal{T}

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\in$ : element of, $\int$ : integral and $\mathcal{T}$ : space of decreasing functions
Referenced by:: §2.6(iii), §2.6(iii)
Permalink:: http://dlmf.nist.gov/2.6.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §2.6(ii), §2.6 and Ch.2

… ►

2.6.13

\left\langle t^{-s-\alpha},\phi\right\rangle=\frac{1}{{\left(\alpha\right)_{s}% }}\int_{0}^{\infty}t^{-\alpha}\phi^{(s)}(t)\,\mathrm{d}t,

\phi\in\mathcal{T}

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\in$ : element of, $\int$ : integral and $\mathcal{T}$ : space of decreasing functions
Referenced by:: §2.6(ii), §2.6(iii)
Permalink:: http://dlmf.nist.gov/2.6.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §2.6(ii), §2.6 and Ch.2

… ►

2.6.14

\left\langle t^{-s-1},\phi\right\rangle=-\frac{1}{s!}\int_{0}^{\infty}(\ln t)% \phi^{(s+1)}(t)\,\mathrm{d}t,

\phi\in\mathcal{T}

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\in$ : element of, $!$ : factorial (as in $n!$ ), $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $\mathcal{T}$ : space of decreasing functions
Referenced by:: §2.6(ii), §2.6(iii)
Permalink:: http://dlmf.nist.gov/2.6.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §2.6(ii), §2.6 and Ch.2

… ►

2.6.16

\left\langle f_{n},\phi\right\rangle=(-1)^{n}\int_{0}^{\infty}f_{n,n}(t)\phi^{% (n)}(t)\,\mathrm{d}t,

\phi\in\mathcal{T}

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\left\langle\NVar{\Lambda},\NVar{\phi}\right\rangle$ : action of distribution on test function, $\in$ : element of, $\int$ : integral, $n$ : nonnegative integer, $f_{n}(t)$ : remainder, $\mathcal{T}$ : space of decreasing functions and $f_{n,n}(t)$ : $n$ th repeated integral
Referenced by:: §2.6(ii)
Permalink:: http://dlmf.nist.gov/2.6.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §2.6(ii), §2.6 and Ch.2

…

4: Software Index

… ► ►►►

	Open Source	With Book	Commercial
…
4.48(v) Testing
…

►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … ►In the list below we identify four main sources of software for computing special functions. … ►

Commercial Software.

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

… ►The following are web-based software repositories with significant holdings in the area of special functions. …

5: Bibliography G

… ►

A. Gervois and H. Navelet (1985a) Integrals of three Bessel functions and Legendre functions. I. J. Math. Phys. 26 (4), pp. 633–644.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt
Referenced by:: §10.22(iv), §10.43(iv).
Encodings:: BibTeX

►

A. Gervois and H. Navelet (1985b) Integrals of three Bessel functions and Legendre functions. II. J. Math. Phys. 26 (4), pp. 645–655.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (B. D. Agrawal), ZentralBlatt
Referenced by:: §10.22(iv), §10.43(iv).
Encodings:: BibTeX

… ►

S. Goldstein (1927) Mathieu functions. Trans. Camb. Philos. Soc. 23, pp. 303–336.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §28.26(i), §28.26(i), §28.8(iii).
Encodings:: BibTeX

… ►

P. Groeneboom and D. R. Truax (2000) A monotonicity property of the power function of multivariate tests. Indag. Math. (N.S.) 11 (2), pp. 209–218.

ⓘ

External Links:: ISSN 0019-3577, Document, MathReview (Sumitra Purkayastha), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

… ►

K. I. Gross and D. St. P. Richards (1991) Hypergeometric functions on complex matrix space. Bull. Amer. Math. Soc. (N.S.) 24 (2), pp. 349–355.

ⓘ

External Links:: ISSN 0273-0979, Document, MathReview, ZentralBlatt
Referenced by:: §35.7(iii), §35.8(i), §35.8(ii), §35.8(iii), §35.8(iv).
Encodings:: BibTeX

…

6: Bibliography B

… ►

H. F. Baker (1995) Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge.

ⓘ

Notes:: Reprint of the 1897 original, with a foreword by Igor Krichever.
External Links:: ISBN 0-521-49877-5, MathReview (John B. Little), ZentralBlatt
Referenced by:: §20.9(ii).
Encodings:: BibTeX

… ►

M. V. Berry and F. J. Wright (1980) Phase-space projection identities for diffraction catastrophes. J. Phys. A 13 (1), pp. 149–160.

ⓘ

External Links:: ISSN 0305-4470, Document, MathReview (J. S. Joel), ZentralBlatt
Referenced by:: §36.9.
Encodings:: BibTeX

… ►

D. K. Bhaumik and S. K. Sarkar (2002) On the power function of the likelihood ratio test for MANOVA. J. Multivariate Anal. 82 (2), pp. 416–421.

ⓘ

External Links:: ISSN 0047-259X, Document, MathReview (Yasuko Chikuse), ZentralBlatt
Referenced by:: §35.9.
Encodings:: BibTeX

… ►

W. Bosma and M.-P. van der Hulst (1990) Faster Primality Testing. In Advances in Cryptology—EUROCRYPT ’89 Proceedings, J.-J. Quisquater and J. Vandewalle (Eds.), Lecture Notes in Computer Science, Vol. 434, New York, pp. 652–656.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §27.18.
Encodings:: BibTeX

… ►

D. M. Bressoud (1989) Factorization and Primality Testing. Springer-Verlag, New York.

ⓘ

External Links:: ISBN 0-387-97040-1, MathReview (Brigitte Vallée), ZentralBlatt
Referenced by:: §23.20(iii), §27.19, Profile David M. Bressoud.
Encodings:: BibTeX

…

7: Bibliography M

… ►

I. G. Macdonald (1990) Hypergeometric Functions.

ⓘ

Notes:: Unpublished Lecture Notes, University of London. An online version exists.
External Links:: FullText
Referenced by:: §35.7(ii), §35.8(iii).
Encodings:: BibTeX

… ►

A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

ⓘ

Notes:: Includes Fortran code, maximum accuracy 20S.
External Links:: ISSN 1017-1398, Document, MathReview, ZentralBlatt
Referenced by:: §6.13, 4th item, §6.21(ii).
Encodings:: BibTeX

… ►

F. Marcellán, M. Alfaro, and M. L. Rezola (1993) Orthogonal polynomials on Sobolev spaces: Old and new directions. J. Comput. Appl. Math. 48 (1-2), pp. 113–131.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (C. M. Joshi), ZentralBlatt
Referenced by:: §18.36(ii), §18.36(ii), Erratum (V1.1.9) for Section 18.36(ii).
Encodings:: BibTeX

… ►

N. W. McLachlan (1934) Loud Speakers: Theory, Performance, Testing and Design. Oxford University Press, New York.

ⓘ

Notes:: Reprinted, with corrections, by Dover Publications Inc., 1960.
External Links:: MathReview, ZentralBlatt
Referenced by:: §11.12.
Encodings:: BibTeX

… ►

J. Morris (1969) Algorithm 346: F-test probabilities [S14]. Comm. ACM 12 (3), pp. 184–185.

ⓘ

Notes:: Includes Algol program, maximum accuracy 12S. For a remark see ACM Trans. Math. Software 14(1988), pp. 288-289
External Links:: ISSN 0001-0782, Document, Remark
Referenced by:: §8.28(iv).
Encodings:: BibTeX

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