convolutions

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1: 30.10 Series and Integrals

… ►For product formulas and convolutions see Connett et al. (1993). …

2: 35.2 Laplace Transform

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Convolution Theorem

►If $g_j$ is the Laplace transform of $f_j$, $j=1,2$, then $g_1{g}_2$ is the Laplace transform of the convolution $f_1\ast f_2$, where ►

35.2.3

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Defines:

$\ast$: convolution operator (locally)

Symbols:

$\int$: integral, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $d$: differential of matrix, : less-than for matrices, $f(𝐗)$: complex-valued function of matrix argument and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.6(ii)

Permalink:

http://dlmf.nist.gov/35.2.E3

Encodings:

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See also:

Annotations for §35.2, §35.2 and Ch.35

3: 27.5 Inversion Formulas

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27.5.1 $$h(n)=\sum _{d|n}f(d)g\left(\frac{n}{d}\right),$$

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Symbols:

$d$: positive integer, $n$: positive integer, $f$: function, $g$: function and $h$: function

Permalink:

http://dlmf.nist.gov/27.5.E1

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See also:

Annotations for §27.5 and Ch.27

►called the Dirichlet product (or convolution) of $f$ and $g$. …

4: 2.6 Distributional Methods

… ►We now derive an asymptotic expansion of ${𝐼}^{\mu }f(x)$ for large positive values of $x$. ►In terms of the convolution product …The replacement of $f(t)$ by its asymptotic expansion (2.6.9), followed by term-by-term integration leads to convolution integrals of the form …However, the left-hand side can be considered as the convolution of the two distributions associated with the functions $t^{\mu -1}$ and $t^{-s-\alpha }$, given by (2.6.12) and (2.6.13). … ►The method of distributions can be further extended to derive asymptotic expansions for convolution integrals: …

5: 37.20 Mathematical Applications

… ►For regular domains, such as square, sphere, ball, simplex, and conic domains, they are used to study convolution structure, maximal functions, and interpolation spaces, as well as localized kernel and localized frames. …

6: 7.21 Physical Applications

… ►Voigt functions $𝖴(x,t)$, $𝖵(x,t)$, can be regarded as the convolution of a Gaussian and a Lorentzian, and appear when the analysis of light (or particulate) absorption (or emission) involves thermal motion effects. …

7: 1.14 Integral Transforms

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Convolution

►For Fourier transforms, the convolution $(f\ast g)(t)$ of two functions $f(t)$ and $g(t)$ defined on $(-\mathrm{\infty },\mathrm{\infty })$ is given by … ►

Convolution

►For Laplace transforms, the convolution of two functions $f(t)$ and $g(t)$, defined on $[0,\mathrm{\infty })$, is … ►

Convolution

…

8: 2.5 Mellin Transform Methods

… ►with . ►One of the two convolution integrals associated with the Mellin transform is of the form ►

2.5.3 $$I(x)={\int }_0^{\mathrm{\infty }}f(t)h(xt)dt,$$ $x>0$,

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $f(x)$: locally integrable function, $I(x)$: convolution integral and $h(x)$: function

Referenced by:

§2.5(ii), §2.5(ii)

Permalink:

http://dlmf.nist.gov/2.5.E3

Encodings:

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See also:

Annotations for §2.5(i), §2.5 and Ch.2

… ►

2.5.29 $$I(x)=\sum _{j,k=1}^2{I}_{jk}(x),$$

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Symbols:

$I(x)$: convolution integral

Referenced by:

§2.5(ii)

Permalink:

http://dlmf.nist.gov/2.5.E29

Encodings:

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See also:

Annotations for §2.5(ii), §2.5 and Ch.2

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2.5.31 $$I_{21}(x)=0,$$ for $x\ge 1$.

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Symbols:

$I(x)$: convolution integral

Permalink:

http://dlmf.nist.gov/2.5.E31

Encodings:

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See also:

Annotations for §2.5(ii), §2.5 and Ch.2

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9: Bibliography G

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Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.

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External Links:

ISSN 0026-9255, MathReview (Peter Stevenhagen), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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Riccati equations and convolution formulae for functions of Rayleigh type. J. Phys. A 33 (7), pp. 1363–1368.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§10.21(xiii).

Encodings:

BibTeX

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10: Bibliography W

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Explicit error terms for asymptotic expansions of Mellin convolutions. J. Math. Anal. Appl. 72 (2), pp. 740–756.

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External Links:

ISSN 0022-247X, Document, MathReview (John S. Lew), ZentralBlatt

Referenced by:

§2.6(ii), §2.6(iv).

Encodings:

BibTeX

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