Intersection bodies and generalized cosine transforms

Abstract

The paper is focused on intimate connection between geometric properties of intersection bodies in convex geometry and generalized cosine transforms in harmonic analysis. A new concept of λ-intersection body, that unifies some known classes of geometric objects, is introduced. A parallel between trace theorems in function theory, restriction onto lower-dimensional subspaces of the spherical Radon transforms and the generalized cosine transforms, and sections of λ-intersection bodies is established. New integral formulas for different classes of cosine transforms are obtained and examples of λ-intersection bodies are given. We also revisit some known facts in this area and give them new simple proofs.

Complex intersection bodies

“…An equivalent and probably more geometric way to define k-intersection bodies would be to say that these bodies are limits in the radial metric of k-intersection bodies of star bodies (see [38] or [41] for a proof of equivalence of this property to the original definition from [25]). …”

Square-like Functions Generated by a Composite Wavelet Transform

“…Landkof [9], M.V. Fedorjuk [5], B.Rubin [11], I.A.Aliev, B.Rubin, S.Sezer and S. Uyhan [3], I.A.Aliev [2]. The following lemma contains some properties of kernels v (β) (y, t) and semigroups V [(see [2], [3])] Let β > 0, t > 0 and y ∈ R n .…”

Bi-Parametric Potentials, Relevant Function Spaces and Wavelet-Like Transforms

“…Note that the β-semigroup (3.1) arises in various contexts of analysis, integral geometry, and probability; see, e.g., [8], [14], [13], [19].…”

“…The positivity of w (β) (y, t) in case of n = 1 can be found in [13, p. 44-45]. The general case n ≥ 1 was investigated by B. Rubin [19], Lemma 7.1. For the statement (c), see [14] when 0 < β ≤ 2, and [8] when 1 < β < ∞.…”

“…For the statement (c), see [14] when 0 < β ≤ 2, and [8] when 1 < β < ∞. See also [19], [2] where it has been established explicit asymtotic formula: …”