Skip to main content
×
Home
    • Aa
    • Aa

Continuity properties of k-plane integrals and Besicovitch sets

  • K. J. Falconer (a1)
Abstract

Let Π be a k-dimensional subspace of Rn(n ≥ 2) and let Π denote its orthogonal complement. If xRn we shall write x = x0 + x with x0 ∈ Πand x ∈ Π . If f(x) is a real measurable function on Rn, the k-plane integral F(Π,x )is defined as the integral of f over the affine subspace Π + x with respect to k-dimensional Lebesgue measure (assuming that the integral exists). If k = 1 we get the x-ray transform that arises in the problem of radiographic reconstruction, and if k = n − 1, the k-plane integral is the usual projection or Radon transform. The paper by Smith, Solmon and Wagner (4) contains a survey of results on k-plane integrals. Here we shall be interested in the behaviour of the F (Π, x ) regarded as a function of x for fixed Π for various classes of function f. We shall obtain some surprisingly strong results on the continuity and differentiability of F(Π,x) with respect to x for almost all Π (in the sense of the appropriate Haar measure). As will be seen the dimensions n and k have a crucial effect on what may be said, and most of our results will be confined to the cases where k > ½n.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

(1)A. S. Besicovitch On Kakeya's problem and a similar one. Math. Z. 27 (1928), 312320.

(3)B. Fisher On a problem of Besicovitch. Amer. Math. Monthly 80 (1975), 785787.

(4)K. T. Smith , D. C. Solmon and S. L. Wagner Practical and mathematical aspects of the problem of reconstructing objects from radiographs. Bull. Amer. Math. Soc. 83 (1977), 12271270.

(5)D. C. Solmon The X-ray transform. J. Math. Anal. Appl. 56 (1976), 6183.

(6)D. C. Solmon A note on k-plane integral transforms. J. Math. Anal. Appl. 71 (1979), 351358.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×