Let $m> n\geq 2$ be two coprime integers. Define $$S:= \left\{\left(\frac{j}{m}, \frac{nj}{m}- \left\lfloor \frac{nj}{m} \right\rfloor\right), j =0,1,..., m-1\right\}\cup\{(1,1)\} \subset [0,1]\times[0,1].$$
Let $f, g : [0,1]\times[0,1]\to \mathbb{R}$ be two real valued functions and define the 'discrete scalar product' $$\langle f, g\rangle_S = \sum_{x\in S} f(x) g(x).$$
I'm trying to check if there is an example of a family of orthogonal two-variate-polynomials with respect to $\langle ., .\rangle_S.$ In the one dimensional case there is the well known Gram polynomials generated using the scalar product $$\sum_{j=0}^m f(j/m) g(j/m),\,\, f,\,g : [0,1]\to \mathbb{R}. $$
Remark: since $n,m$ are coprime, we have $\{j/m, \, j = 0,.., m-1\}=\left\{\frac{nj}{m}- \left\lfloor \frac{nj}{m} \right\rfloor,\, j = 0,.., m-1\right\}.$
Thank you for any hint.