General Solution of Convolution Equation in Domains with Spherical Symmetry

V. V. Volchkov
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V. V. Volchkov: Donetsk National University, Department of Mathematics

Chapter Chapter 2 in Integral Geometry and Convolution Equations, 2003, pp 169-190 from Springer

Abstract: 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 2.1 $$ \tilde \phi \left( z \right) = \left\langle {\phi ,J_{\left( {n/2} \right) - 1} \left( {z\left| x \right|} \right)\left( {z\left| x \right|^{1 - \left( {n/2} \right)} } \right)} \right\rangle . $$ Since ϕ is radial, for any f ∈ ∊ (ℝ n we have 2.2 $$ \left\langle {\phi ,f} \right\rangle = \left\langle {\phi ,\int\limits_{SO\left( n \right)} {f\left( {\tau x} \right)d\tau } } \right\rangle = \frac{{\left\langle {\phi ,f_{0,1} \left( \rho \right)} \right\rangle }} {{\sqrt {\omega _{n - 1} } }} $$ (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 2.3 $$ \left\langle {\phi ,e^{iz\left( {x,\xi } \right)} } \right\rangle = \frac{{\left( {2\pi } \right)^{n/2} }} {{\omega _{n - 1} }}\tilde \phi \left( z \right) $$ (see also (1.5.29)).

Keywords: General Solution; Induction Hypothesis; Spherical Symmetry; Radial Function; Closed Ball (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1007/978-94-010-0023-9_15

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