Estimating the number of interpolation points for Gaussian-type cubature formula (Q1176933)

Let \(\{L;m\}\) be the set of cubatures \(L_ nf=\sum^ N_{j=1}c_ jf(x_ 1^{(j)},x_ 2^{(j)},x_ 3^{(j)})\) which are exact on the trigonometric polynomials \[ t(x_ 1,x_ 2,x_ 3)=a_ 0+\sum_{0<| n|\leq m}a_ n\exp i(n_ 1x_ 1+n_ 2x_ 2+n_ 3x_ 3),\quad | n|=| n_ 1|+| n_ 2|+| n_ 3|. \] Denote \(N_{\min}(m)=m/n\{N:L_ N\in\{L;m\}\}\), \(L_{N_{\min}}(m)\) is the Gauss cubature. The author constructs a cubature (rather complicated to be reproduced here) for which he proves that \(\in\{L;m\}\). For the number of knots of the Gauss cubature it is obtained an upper estimate.