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 
.