Application of Banach measure to quadrature formulas (Q1821285)

If \(f\in W(r)\), i.e., \(f(t)=f(t_ 1,...,t_ s)\) is a function of the form \(f(t)=c_ 0+\sum_{n\in Z^ s_ 0}c_ ne^{2\pi i(n,t)},\) \(c_ n=a_ n+ib_ n\), with \(\sum_{(n_ 1,...,n_ s)\in Z^ s_ 0}| c_{n_ 1,...,n_ s}|^ 2\overline{(2\pi n_ 1)}^{2r_ 1}...\overline{2\pi n_ s)}^{2r_ s}\leq 1,\) where \(\bar x = \max\{| x|,1\}\), then the author investigates an equality of the following kind \[ \int_{W(r)}| \sum^{N}_{k=1}\rho_ kf(\sigma_ k)-\int_{[0,1]^ s}f(x)dx|^ 2d\mu_{r,\Phi,\Gamma}(f)= \] \[ =\sum^{\infty}_{m=1}\sum_{n\in \Gamma_ m\setminus \Gamma_{m-1}}| L_ n(\rho,\sigma)|^ 2\int...\int_{D_ m}(a^ 2_ n+b^ 2_ n)\phi_ m. \]