Integral Geometry and Convolution Equations pp 169-190 | Cite as
General Solution of Convolution Equation in Domains with Spherical Symmetry
Chapter
Abstract
For the rest of Part 3 we assume that n ⩾ 2. Let ϕ ∈ ɛ′rad(ℝ n ), ϕ ≠ 0. The spherical transform \(
\tilde \phi :\mathbb{C} \to \mathbb{C}
\) of the distribution ϕ is defined by the equality Since ϕ is radial, for any f ∈ ∊ (ℝ n we have (see (1.2.3) and (1.5.10)). In particular, setting f(x) = e iz(x, ξ) , where z ∈ ℂ, \(
\xi \in \mathbb{S}^{n - 1}
\), from (2.2) and (2.1) we obtain (see also (1.5.29)).
$$
\tilde \phi \left( z \right) = \left\langle {\phi ,J_{\left( {n/2} \right) - 1} \left( {z\left| x \right|} \right)\left( {z\left| x \right|^{1 - \left( {n/2} \right)} } \right)} \right\rangle .
$$
(2.1)
$$
\left\langle {\phi ,f} \right\rangle = \left\langle {\phi ,\int\limits_{SO\left( n \right)} {f\left( {\tau x} \right)d\tau } } \right\rangle = \frac{{\left\langle {\phi ,f_{0,1} \left( \rho \right)} \right\rangle }}
{{\sqrt {\omega _{n - 1} } }}
$$
(2.2)
$$
\left\langle {\phi ,e^{iz\left( {x,\xi } \right)} } \right\rangle = \frac{{\left( {2\pi } \right)^{n/2} }}
{{\omega _{n - 1} }}\tilde \phi \left( z \right)
$$
(2.3)
Keywords
General Solution Induction Hypothesis Spherical Symmetry Radial Function Closed Ball
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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© Springer Science+Business Media Dordrecht 2003