There are well known Turan and Delsarte extremal problems for positive definite functions. They are used in analytic number theory, digital signal processing, in estimates of cardinality of codes, designs, contact numbers, the packing density of homogeneous spaces. In these problems is sought the greatest mean value of a positive definite function with a fixed value at zero and given support (Turan problem) or non-positive outside of a given set (Delsarte problem). Turan problem for one-dimensional torus and support on the segment with rational endpoints is reduced to a discrete version of the well-known Fejér extremal problem about the greatest value of non-negative trigonometric polynomial with fixed mean value. Turan and Delsarte problems for symmetric association schemes are discrete extremal problems directly. In this paper, a solution of discrete Fejér, Turan and Delsarte extremal problems, discrete Jackson l₂-constants problem on cyclic group Z_q and direct product of finite cyclic groups is given. The maximal cardinalities of codes for two symmetric association schemes on direct product of finite cyclic groups are calculated. They have provided two-sided estimates in multidimensional discrete Fejér, Turan and Delsarte extremal problems.

finite cyclic group, discrete trigonometric polynomial, symmetric association scheme, code, Fejér problem, Turan problem, Delsarte problem

Mathematical problems and theory of numerical methods