The inversion of Abel's integral equation in astrophysical problems.
Abstract
The inversion of Abel's integral equation is discussed. It is established that the problem of inferring the source function from typical data is illposed, thus implying that small measurement errors severely distort the numerical solution. Since the measured function carries only rudimentary information on the high frequency components of the unknown distribution, it follows that in practice only the first few Fourier coefficients of the solution (typically smaller than 5) can be determined by employing classical inversion techniques. To extract more detailed information, the problem must be reformulated to exploit any a priori knowledge of the solution. Numerical examples are used to illustrate how the inversion can be dramatically improved by invoking, in a very simple and natural way, information on the local smoothness of the source function. It is strongly recommended that similar nonclassical techniques be adopted in practical inversion problems.
 Publication:

Astronomy and Astrophysics
 Pub Date:
 October 1979
 Bibcode:
 1979A&A....79..121C
 Keywords:

 Abel Function;
 Applications Of Mathematics;
 Astrophysics;
 Integral Equations;
 Background Noise;
 Data Acquisition;
 Data Processing;
 Fourier Transformation;
 Inference;
 Instrument Errors;
 Inversions;
 Laplace Transformation;
 Numerical Analysis;
 Problem Solving;
 Astrophysics;
 Mathematics;
 Computing;
 Data Processing