The following theorem links the Fejér-problem directly with the existence problem of multivariate Gauss-quadratures and in case of with the existence of a 2m-design.
Thus, for , there exist no Gauss-quadratures for and for , they do not exist for . There might be Gauss-quadratures for m=2. But even in these cases it turns out that certain restrictions have to hold.
That is, existence is possible for and it is known for .