Summary: This is the third article in a series dedicated to Smolyak grids. The work relates to analytical number theory and it deals with the application of number theory to problems of approximate analysis. The paper shows that:

1

the linear operator \(A_q\) of weighted grid averages over the Smolyak grid at dimension \(s\ge3\) is not normal;

2

found the values of some trigonometric sums \(S_q(m_1,\ldots,m_s)\) of the resin grid at the dimension \(s\ge3\).