Die Struktur rotationsinvarianter Paley-Wiener-Räume : mit einer Anwendung auf Abtastprobleme
The Paley-Wiener classes belong to the classical function spaces, which are used to model one and multi-dimensional signals. They consist of functions which are defined on some finite-dimensional real Euclidean space and whose Fourier transforms are supported in a fixed and in general compact subset of this Euclidean space. Here we are interested in the structure of rotation invariant Paley-Wiener spaces, i.e. in Paley-Wiener spaces which are invariant under the action of the special orthogonal group. The main tool in this connection is the Fourier-Laplace expansion. The rotation invariant Paley-Wiener spaces are completely characterized in terms of the so called radial Fourier-Laplace coefficients. In this context we in particular derive a Paley-Wiener-type theorem for certain Bessel-Hankel transforms. Besides the characterization of the rotation invariant Paley-Wiener spaces by means of the radial Fourier-Laplace coefficients we obtain an orthogonal decomposition of these spaces into closed and rotation invariant subspaces and (again with the aid of the Fourier-Laplace expansion) a linear approximation process in these spaces. This process approximates in the quadratic mean and uniformly simultaneously. For the classes of signals to be approximated we finally derive some families of sampling expansions, which include and generalize some methods used in the technical literature.
info:eu-repo/classification/ddc/510, Mathematik, Paley-Wiener-Räume, Fourier-Laplace-Entwicklung, Bessel-Hankel-Transformationen, Paley-Wiener-Typ Sätze, Sampling, Quasi-Sampling
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