Generalized Fourier transform on an arbitrary triangular domain
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Springer 2005
Generalized Fourier transform on an arbitrary triangular domain ∗ Jiachang Sun and Huiyuan Li Institute of Software, Chinese Academy of Sciences, Beijing, 100080, P.R. China E-mail: {sun;hynli}@mail.rdcps.ac.cn
Received 11 July 2002; accepted 21 July 2003 Communicated by Y. Xu
In this paper, we construct generalized Fourier transform on an arbitrary triangular domain via barycentric coordinates and PDE approach. We start with a second-order elliptic differential operator for an arbitrary triangle which has the so-called generalized sine (TSin) and generalized cosine (TCos) systems as eigenfunctions. The orthogonality and completeness of the systems are then proved. Some essential convergence properties of the generalized Fourier series are discussed. Error estimates are obtained in Sobolev norms. Especially, the generalized Fourier transforms for some elementary polynomials and their convergence are investigated. Keywords: arbitrary triangle, eigen-decomposition, generalized Fourier transform, generalized Fourier series, convergence, error estimates
1.
Introduction
It is well known that the concept of Fourier transform plays a key role in numerical analysis. There has been a lot of work relating to Fourier series, Fourier integral, finite Fourier transform, FFT (Fast Fourier Transform) and so on. As we know, the original result has first been studied in the univariate case and then been extended into multivariate cases by using tensor product. Rigorously, the tensor product approach is still staying in the one dimensional level via decreasing dimension, and its result can only be used in box domains. Recently, there have been increasing interests in how to extend Fourier methods into irregular domains in high dimension. Since triangle, tetrahedron and generally simplex are the simplest non-box domains, more and more attention has being paid to the studies on these domains [2,5,7,9,11,12,16,18–22]. The basic issue of the generalization of Fourier transform for non-rectangle domains is the construction of the orthogonal bases. There are many different approaches ∗ This work was supported by the Major Basic Project of China (No. G19990328) and National Natural
Science Foundation of China (No. 60173021).
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in literature for obtaining explicit representations of orthogonal bases. The first method is based on the orthogonalization of some known non-orthogonal (usually biorthogonal or semi-orthogonal) bases such as Appell’s polynomials [1,4,8]. The second approach is obtained from a transformation of the triangle or simplex to the square domain [5,6,9]. We point out that PDE approach is one of the most efficient methods in constructing orthogonal bases. As is well known, the eigen-decompositions of Laplacian and singular Sturm–Liouville operator usually result in the orthogonal trigonometric and polynomial bases on box domains, respectively. It is reasonable as well to expect the eigendecompositions of some certain differential operators to construct
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