# convolutions

(0.001 seconds)

## 1—10 of 14 matching pages

##### 1: 30.10 Series and Integrals
For product formulas and convolutions see Connett et al. (1993). …
##### 2: 35.2 Laplace Transform
###### 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 $f_{1}*f_{2}$, where
##### 3: 27.5 Inversion Formulas
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: 7.21 Physical Applications
Voigt functions $\mathsf{U}\left(x,t\right)$, $\mathsf{V}\left(x,t\right)$, 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. …
##### 6: 1.14 Integral Transforms
###### Convolution
For Fourier transforms, the convolution $(f*g)(t)$ of two functions $f(t)$ and $g(t)$ defined on $(-\infty,\infty)$ is given by …
###### Convolution
For Laplace transforms, the convolution of two functions $f(t)$ and $g(t)$, defined on $[0,\infty)$, is …
##### 7: 2.5 Mellin Transform Methods
with $a. One of the two convolution integrals associated with the Mellin transform is of the form
2.5.29 $I(x)=\sum\limits_{j,k=1}^{2}I_{jk}(x),$
2.5.31 $I_{21}(x)=0,$ for $x\geq 1$.
##### 8: Bibliography G
• K. Girstmair (1990b) Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.
• D. P. Gupta and M. E. Muldoon (2000) Riccati equations and convolution formulae for functions of Rayleigh type. J. Phys. A 33 (7), pp. 1363–1368.
• ##### 9: Bibliography W
• R. Wong (1979) Explicit error terms for asymptotic expansions of Mellin convolutions. J. Math. Anal. Appl. 72 (2), pp. 740–756.
• ##### 10: Bibliography
• W. L. Anderson (1982) Algorithm 588. Fast Hankel transforms using related and lagged convolutions. ACM Trans. Math. Software 8 (4), pp. 369–370.