Volume 24, Issue 3
Multivariate Fourier Transform Methods Over Simplex and Super-Simplex Domains
DOI:

J. Comp. Math., 24 (2006), pp. 305-322.

Published online: 2006-06

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• Abstract

In this paper we propose the well-known Fourier method on some non-tensor product domains in $\textrm{R}^d$, 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 Laplacian-like operator over these domains is explored.

• Keywords

Multivariate Fourier transform Simplex and super-simplex Multivariate sine and cosine functions Eigen-decomposition for Laplacian-like operator Multivariate fast Fourier transform

@Article{JCM-24-305, author = {}, title = {Multivariate Fourier Transform Methods Over Simplex and Super-Simplex Domains}, journal = {Journal of Computational Mathematics}, year = {2006}, volume = {24}, number = {3}, pages = {305--322}, abstract = { In this paper we propose the well-known Fourier method on some non-tensor product domains in $\textrm{R}^d$, 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 Laplacian-like operator over these domains is explored. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8754.html} }
TY - JOUR T1 - Multivariate Fourier Transform Methods Over Simplex and Super-Simplex Domains JO - Journal of Computational Mathematics VL - 3 SP - 305 EP - 322 PY - 2006 DA - 2006/06 SN - 24 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8754.html KW - Multivariate Fourier transform KW - Simplex and super-simplex KW - Multivariate sine and cosine functions KW - Eigen-decomposition for Laplacian-like operator KW - Multivariate fast Fourier transform AB - In this paper we propose the well-known Fourier method on some non-tensor product domains in $\textrm{R}^d$, 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 Laplacian-like operator over these domains is explored.