On cardinality of the lower sets and universal discretization
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A set Q in ℤ_+^d is a lower set if (k_1,…,k_d)∈ Q implies (l_1,…,l_d)∈ Q whenever 0≤ l_i≤ k_i for all i. We derive new and refine known results regarding the cardinality of the lower sets of size n in ℤ_+^d. Next we apply these results for universal discretization of the L_2-norm of elements from n-dimensional subspaces of trigonometric polynomials generated by lower sets.
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