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Dyadic distributions
B. I. Golubov Moscow Engineering Physics Institute (State University)
Abstract:
On the basis of the concept of pointwise dyadic derivative dyadic distributions are introduced as continuous linear functionals on the linear space
$D_d(\mathbb R_+)$ of infinitely differentiable functions compactly supported by the positive half-axis
$\mathbb R_+$
together with all dyadic derivatives. The completeness of the space
$D'_d(\mathbb R_+)$ of dyadic distributions is established. It is shown that a locally
integrable function on
$\mathbb R_+$ generates a dyadic
distribution.
In addition, the space
$S_d(\mathbb R_+)$
of infinitely dyadically differentiable
functions on
$\mathbb R_+$ rapidly decreasing
in the neighbourhood of
$+\infty$ is defined. The space
$S'_d(\mathbb R_+)$ of dyadic distributions of slow growth
is introduced as the space of continuous linear functionals
on
$S_d(\mathbb R_+)$. The completeness of the space
$S'_d(\mathbb R_+)$ is established; it is proved that each integrable
function on
$\mathbb R_+$ with polynomial
growth at
$+\infty$ generates
a dyadic distribution of slow growth.
Bibliography: 25 titles.
UDC:
517.982.4
MSC: 46F05,
42C10 Received: 18.04.2005 and 30.10.2006
DOI:
10.4213/sm3780
Fulltext:
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