Book contents
- Frontmatter
- Contents
- Preface
- Glossary of Selected Symbols
- 1 Introduction
- 2 Smoothness
- 3 Uniqueness
- 4 Identifying Functions and Directions
- 5 Polynomial Ridge Functions
- 6 Density and Representation
- 7 Closure
- 8 Existence and Characterization of Best Approximations
- 9 Approximation Algorithms
- 10 Integral Representations
- 11 Interpolation at Points
- 12 Interpolation on Lines
- References
- Supplemental References
- Author Index
- Subject Index
10 - Integral Representations
Published online by Cambridge University Press: 05 August 2015
- Frontmatter
- Contents
- Preface
- Glossary of Selected Symbols
- 1 Introduction
- 2 Smoothness
- 3 Uniqueness
- 4 Identifying Functions and Directions
- 5 Polynomial Ridge Functions
- 6 Density and Representation
- 7 Closure
- 8 Existence and Characterization of Best Approximations
- 9 Approximation Algorithms
- 10 Integral Representations
- 11 Interpolation at Points
- 12 Interpolation on Lines
- References
- Supplemental References
- Author Index
- Subject Index
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 ψw := eix·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 Δx denote the Laplacian with respect to the variables x1, …, xn, 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].
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- Ridge Functions , pp. 141 - 151Publisher: Cambridge University PressPrint publication year: 2015