In this paper we propose the well-known Fourier method on some non-tensor product domains in Rd, including simplex and so-called super-simplex which consists of （d ＋ 1）! simplices. As two examples, in 2-D and 3-D c...In this paper we propose the well-known Fourier method on some non-tensor product domains in Rd, including simplex and so-called super-simplex which consists of （d ＋ 1）! simplices. As two examples, in 2-D and 3-D case a super-simplex is shown as a parallel hexagon and a parallel quadrilateral dodecahedron, respectively. We have extended most of concepts and results of the traditional Fourier methods on multivariate cases, such as Fourier basis system, Fourier series, discrete Fourier transform （DFT） and its fast algorithm （FFT） on the super-simplex, as well as generalized sine and cosine transforms （DST, DCT） and related fast algorithms over a simplex. The relationship between the basic orthogonal system and eigen-functions of a LaDlacian-like operator over these domains is explored.展开更多
基金This work was partly supported by National Science Foundation of China （No. 10431050 and 60573023）, the Major Basic Project of China （2005CB321702） and by Natural Science Foundation of United States （No. CCF0305666） during the author＇s visit at University of Colorado at Boulder.
文摘In this paper we propose the well-known Fourier method on some non-tensor product domains in Rd, including simplex and so-called super-simplex which consists of （d ＋ 1）! simplices. As two examples, in 2-D and 3-D case a super-simplex is shown as a parallel hexagon and a parallel quadrilateral dodecahedron, respectively. We have extended most of concepts and results of the traditional Fourier methods on multivariate cases, such as Fourier basis system, Fourier series, discrete Fourier transform （DFT） and its fast algorithm （FFT） on the super-simplex, as well as generalized sine and cosine transforms （DST, DCT） and related fast algorithms over a simplex. The relationship between the basic orthogonal system and eigen-functions of a LaDlacian-like operator over these domains is explored.
