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Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition
book

Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition

by Andrei D. Polyanin, Vladimir E. Nazaikinskii
December 2015
Intermediate to advanced content levelIntermediate to advanced
1643 pages
59h 33m
English
Chapman and Hall/CRC
Content preview from Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd Edition
1270 ELEMENTS OF THEORY OF GENERALIZED FUNCTIONS
21.1.6 Tempered Distributions and Fo u rier Transform
Tempered distributions.
The space J of rapidly decaying test functions contains all infinitely differentiable func-
tions ϕ(x) decaying, together with all of their derivatives, more rapidly than any power of
|x|
1
as |x| . The convergence in J is defined as follows: a sequence of functions
ϕ
n
J converges to a function ϕ J if and only if x
β
d
m
dx
m
ϕ
n
(x) x
β
d
m
dx
m
ϕ(x) as
n for any β and m = 0, 1, . . . .
We have the following consequences of these definitions: (i) J is a linear space; (ii)
K J; (iii) the convergence in K implies the convergence ...
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Publisher Resources

ISBN: 9781466581494