Parallelepipedal cubature formulas and perfect splines (Q805961)

The author considers classes of periodic differentiable functions of two variables whose rth \((r=3,4)\) differential exists a.e. and its norm is bounded. He introduces the so-called parallelepiped cubature formulae with equal weights and nodes lying on a two-dimensional lattice consistent with the periodicity of the function. Two theorems are presented stating that for \(r=3\) the formulae with triangular lattices of nodes are better than those with rectangular lattices whereas for \(r=4\) this is not true. The proof is given only for \(r=3\). The author often refers the reader to his previous paper on the same topic [Mat. Zametki 45, No.4, 121-124 (1989)] where, however, no proofs of the statements were given.