10 - Integral Representations

Summary

In this chapter we consider integral representations of functions using kernels that are ridge functions. The most commonly used integral representation with a ridge function kernel is that given by the Fourier transform. The function ψ_w := e^ix·w is, for each fixed w, a ridge function. Under suitable assumptions on an f defined on Rⁿ, we have

For example, the above holds if f ∊ L¹(Rⁿ) n C(ⁿ) and,. While we have drifted into the complex plane, we can easily rewrite the above using only real-valued functions.

Another integral representation with a ridge function kernel may be found in John [1955], p. 11. We present it here without proof. Assume, i.e., f and its first partial derivatives are continuous functions, and f has compact support. Let Δ_x denote the Laplacian with respect to the variables x₁, …, x_n, i.e.,

Let Sⁿ⁻¹ denote the unit sphere in Rⁿ, and let da be uniform measure on Sⁿ⁻¹ of total measure equal to the surface area of the unit sphere Sⁿ⁻¹, i.e., of total measure

For n and k odd positive integers, we have

For n an even positive integer and k any even non-negative integer, we have

The relationship between these integral formulæ and the inverse Radon transform can be found in John [1955], together with other similar integral representations.

In this chapter we review two additional integral representations. The first, using an orthogonal decomposition in terms of Gegenbauer polynomials is from Petrushev [1998]. The second is based upon ridgelets and was presented by Candès [1998] in his doctoral thesis, see also Candès [1999].