Necessary Conditions for Existence of Some Designs in Polynomial Metric Spaces

Let M be a metric space with a finite diameter D and a finite normalized measure μM. Let the Hilbert space L2(M) of complex-valued functions be decomposed into a countable (whenM is infinite) or finite (with D+ 1 members whenM is finite) direct sum of mutually orthogonal subspaces L2(M) = V0 ⊕ V1 ⊕ · · · (V0 is the space of constant functions). We denote N = D + 1 ifM is finite and N = ∞ otherwise. In definition and description of the notion of polynomial metric spaces we follow [26, 27].