TY - JOUR
T1 - Multivariate Fourier Transform Methods over Simplex and Super-Simplex Domains
AU - Jiachang Sun
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, Simplex and super-simplex, Multivariate sine
and cosine functions, Eigen-decomposition for Laplacian-like operator, Multivariate fast
Fourier transform.
AB - <p style="text-align: justify;">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.</p>