What is a tensor-product Chebyshev grid?

Question

What is the difference between "Chebyshev grid" and "tensor-product Chebyshev grid"? Are they defined on a 2D vector?

Answer 1

Chebyshev grid is a set of points on an interval. For example, on the interval $[-1,1]$ the Chebyshev grid $C_n$ consists of the points $\cos\left(\frac{2k-1}{2n}\pi\right)$, $k=1,\dots,n$. It looks like this (points placed above the axis for better visibility):

A tensor product grid in 2D is a set of the form $C_n\times C_m = \{(x,y): x\in C_n, y\in C_m\}$ where $C_n, C_m$ are one-dimensional grids. This is nothing but the Cartesian product of two sets. It can be similarly defined in higher dimensions. (As a practical point, in high dimensions the number of points can become very large, motivating the search for sparse grids.)

Here is the Chebyshev grid $C_{10}\times C_{10}$ on $[-1,1]\times [-1,1]$.