A lower bound for n-diameters

Let D be a subset of the s-dimensional lattice Z^S, M=M(D) the number of elements in D, _Dthe space of trigonometric polynomials on the torus T^S with spectrum concentrated in D and having unit norm in L₂(T^S). In this paper we give the following bound for the Gel'fand diameter:d ⁿ( _D,C(T^s))M/2–N/2. This bound is subsequently used for actual functional classes.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 413–419, October, 1972.