离散余弦变换的连续Fourier延拓性质及其多维推广
Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization
摘要：
本文给出了一种计算连续函数$g(t)$的离散傅里叶变换（DFT）的通用方法，该连续函数由其在紧致半单李群的基本区域$F$中的有限共轭阶元素的共轭类所生成的均匀网格$F_{N}$上的点处的值$g_{j}$给出.当$F$被简化为一维线段时，本方法的实现是针对SU（2）群的，而对于$SU(2)\times ... \times SU(2)$在多维情况下。这个最简单的情况导致了一种称为离散余弦变换（DCT）的变换，它通常被认为只是标准DFT的一种特定类型。在这里，我们表明，DCT是非常不同的标准DFT时，这两个离散变换的连续扩展的属性，从离散网格点$t_j; j=0,1, ... N$的所有点$t \in F$被认为是。(A)与DFT的连续扩展不同，（逆）DCT的连续扩展（称为CEDCT）非常接近网格点$t_j$之间的$g(t)$。(B)对于增加的$N$，CEDCT的导数收敛到$g(t)$的导数。（C）对于CEDCT，局部性原理成立。最后，我们使用二维DCT的连续扩展来说明它的插值潜力，以及二维图像的数据压缩。
英文摘要：
A versatile method is described for the practical computation of the discrete Fourier transforms (DFT) of a continuous function $g(t)$ given by its values $g_{j}$ at the points of a uniform grid $F_{N}$ generated by conjugacy classes of elements of finite adjoint order $N$ in the fundamental region $F$ of compact semisimple Lie groups. The present implementation of the method is for the groups SU(2), when $F$ is reduced to a one-dimensional segment, and for $SU(2)\times ... \times SU(2)$ in multidimensional cases. This simplest case turns out to result in a transform known as discrete cosine transform (DCT), which is often considered to be simply a specific type of the standard DFT. Here we show that the DCT is very different from the standard DFT when the properties of the continuous extensions of these two discrete transforms from the discrete grid points $t_j; j=0,1, ... N$ to all points $t \in F$ are considered. (A) Unlike the continuous extension of the DFT, the continuous extension of (the inverse) DCT, called CEDCT, closely approximates $g(t)$ between the grid points $t_j$. (B) For increasing $N$, the derivative of CEDCT converges to the derivative of $g(t)$. And (C), for CEDCT the principle of locality is valid. Finally, we use the continuous extension of 2-dimensional DCT to illustrate its potential for interpolation, as well as for the data compression of 2D images.
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