By Niçaise

Nicaise S. examine numerique et equations aux derivees partielles (Dunod, 2000)(fr)(ISBN 2100049410)

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**Sample text**

4. Integral of a Function over a Given Region in Terms of Integrals over Hyperplanes Consider the following problem. Let φ(ξ, p) be the integrals of some f(x) over all possible hyperplanes. We wish to calculate the integral of f(x) over a given region V. To solve this problem we introduce the characteristic function χ(χ) of V. We may then write j f(x) dx = J x(x)f(x) dx. Let S(£, p) be the Radon transform of χ(χ). Then from the definition of the Radon transform of a generalized function we have j x{x)f(x) dx = {~^li j r [ £ S(f, Pfoir^f,P) dp] ω(ξ).

3a. , xn) for Nonnegative Integer k The result we have just obtained applies for noninteger λ. The limit of integer λ illustrates an instructive point, and we will therefore go into it in some detail. 8 to calculate f χλ dx. J0 28 RADON TRANSFORM OF TEST FUNCTIONS Ch. , where (£ι)+ 1-λ and (£ι)ΐ 1-Λ also have poles. , xn) is not itself singular at these values of λ. It will be seen that there is, however, no contradiction when it is recalled that the Radon transform of a generalized function is defined only to within an unessential function.

Each of these integrals converges for Re λ > — 1, and each can be continued analytically in λ to Re λ < — 1. 6. Radon Transform of the Characteristic Function of One Sheet of a Two-Sheeted Hyperboloid In this section we shall derive the expression for the Radon transform of the characteristic function of one sheet of the two-sheeted hyperboloid, that is, of the region defined by 4-4-4>i- *i>o. 6 The Radon Transform of Generalized Functions 41 There are two types of plane sections through this hyperboloid, namely ellipses and hyperbolas, and the Radon transform of its characteristic function should contain two terms, one of which is the area of an ellipse, and the other the area of a hyperbola.