Abstract
If in spherical stereology the actual radius of spheres obeys a discrete probability distribution with unknown jump points the solution of the relevant Abel integral equation is not continuous, and hence the supremum norm is inappropriate for estimating the error of an approximation. We show that the backward Euler method converges in the L1-norm, also in the more general case that the distribution function is a linear combination of Heaviside functions superimposed on a Lipschitz-continuous background. We treat both cases:
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(a)
cutting plane (first kind integral equation),
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(b)
cutting slice (second kind integral equation).
For convenience (in order to avoid a superficial nonlinearity) we work with number densities and corresponding cumulative functions instead of with probability densities and distribution functions.
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References
R.S. Anderssen and A.J. Jakeman: Product integration for particle size distributions. Utilitas Mathematica 8 (1975), 111–126.
G. Bach: Über die Größenverteilung von Kugelschnitten in durchsichtigen Schnitten endlicher Dicke. Zeitschrift für wissenschaftliche Mikroskopie 64 (1959), 265–270.
V.L. Bakke and Z. Jackiewicz: Stability analysis of product 0-methods for Abel integral equations of the second kind. Num. Math. 48 (1986), 127–136.
N.G. de Bruijn and P. Erdös: On a recursion formula and on some Tauberian theorems. Journal of Research of the National Bureau of Standards 50 (1953), 161–164.
R.F. Cameron and S. McKee: The analysis of product integration methods for Abel’s equation using discrete fractional differentiation. IMA Journal of Numerical Analysis 5 (1985), 339–353.
P.P.B. Eggermont: A new analysis of the Euler-, midpoint-and trapezoidal-discretization methods for the numerical solution of Abel-type integral equations. Medical Image Processing Group Technical Report No. MIPG34 September 1979. State University of New York at Buffalo, Department of Computer Science.
P.P.B. Eggermont: A new analysis of the trapezoidal-discretization method for the numerical solution of Abel-type integral equations. Journal of Integral Equations 3 (1981), 317–332.
P. Erdös, W. Feller, and H. Pollard: A property of power series with positive coefficients. Bulletin of the American Mathematical Society 55 (1949), 201–204.
R.L. Fullman: Measurement of particle sizes in opaque bodies. Journal of Metals 5 (1953), 447–452.
P.L. Goldsmith: The calculation of true particle size distributions from the sizes observed in a thin slice. British Journal of Applied Physics 18 (1967), 813–830.
R. Gorenflo: How to compute very rough solutions of Abel integral equations? International Series of Numerical Mathematics 73 (1985), 281–282.
R. Gorenflo: Abel integral equations: applications-motivated solution concepts. To appear in Methoden und Verfahren der Mathematischen Physik.
W.C. Krumbein: Thin section mechanical analysis of inducted sediments. Journal of Geology 43 (1935), 482–496.
W.P. Reid: Distribution of sizes of spheres in a solid from a study of slices of the solid. Journal of Mathematics and Physics 34 (1955), 95–102.
E. Scheil: Die Berechnung der Anzahl und Größenverteilung kugelförmiger Kristalle in undurchsichtigen Körpern mit Hilfe der durch einen ebenen Schnitt erhaltenen Schnittkreise. Zeitschrift für anorganische und allgemeine Chemie 201 (1931), 259–264.
S.D. Wickseil: The corpuscle problem. A mathematical study of a biometric problem. Biometrika 17 (1925), 84–99.
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© 1986 Birkhäuser Verlag Basel
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Gorenflo, R. (1986). Approximation of Discrete Probability Distributions in Spherical Stereology. In: Cannon, J.R., Hornung, U. (eds) Inverse Problems. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 77. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-7014-6_7
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DOI: https://doi.org/10.1007/978-3-0348-7014-6_7
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