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Orthonormal Bases

  • Volker Michel
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

Within Part III, we consider functions on the closed ball
$$\mathcal{B} := \left\{\left.x \in {\mathbb{R}^3}\,\right\vert \,\vert x\vert \leq \beta \right\}$$
(9.1)
with the radius \(\beta > 0\). Typical applications, where such functions have to be approximated, are tomographic problems in geophysics or medical imaging. Since the Earth’s interior and the human brain do—roughly speaking—consist of layers with concentric spheres as boundaries, a tensor product ansatz for the Cartesian coordinates \(x,y\), and \(z\) (see Sect.3.5) is not useful here. Instead, a separation \(x = r\xi \), \(r = \vert x\vert \epsilon \mathbb{R}_{0}^+\), \(\xi \in \Omega \), is more appropriate for practical purposes.

Keywords

Orthonormal System Radial Part Algebraic Polynomial Angular Part Concentric Sphere 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Volker Michel
    • 1
  1. 1.Geomathematics Group Department of MathematicsUniversity of SiegenSiegenGermany

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