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On the null space of a class of Fredholm integral equations of the first kind

  • Volker Michel and Sarah Orzlowski EMAIL logo
Published/Copyright: September 1, 2015

Abstract

We investigate the null space of Fredholm integral operators of the first kind with

TD:=D(x)k(x,)dx,

where is a ball, the integral kernel satisfies

k(x,y)=n=0cn|x|ln|y|n-2+qPn(q)(x|x|y|y|),x,yq,

where (cn) and (ln) are sequences with particular constraints, and the Pn(q) are Gegenbauer polynomials. We first discuss the case of a 3-dimensional ball in detail, where the Pn(3)=Pn are Legendre polynomials, and then derive generalizations for the q-dimensional ball. The discussed class includes some important tomographic inverse problems in the geosciences and in medical imaging. Amongst others, uniqueness constraints are proposed and compared. One result is that information on the radial dependence of D is lost in TD. We are also able to generalize a famous result on the null space of Newton’s gravitational potential operator to the q. Moreover, we characterize the orthonormal basis of the derived singular value decomposition of T as eigenfunctions of a differential operator and as basis functions of a particular Sobolev space. Our results give further insight to the interconnections of magnetic field inversion on the one side and gravitational/electric field inversion on the other side.

Award Identifier / Grant number: MI 655/10-1

Funding statement: The research was supported by DFG MI 655/10-1.

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Received: 2015-3-3
Revised: 2015-7-17
Accepted: 2015-7-26
Published Online: 2015-9-1
Published in Print: 2016-12-1

© 2016 by De Gruyter

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