Explicit uniformly bounded bases of homogeneous holomorphic polynomials in every dimension
We construct uniformly bounded orthonormal bases for homogeneous holomorphic polynomials on complex unit spheres. Fix d >= 2 and N >= 0, and put m = d-1 and L = N+d-1. For each multi-index alpha = (alpha_1,...,alpha_d) of nonnegative integers with sum N, define s_b(alpha) = b + alpha_1 + ... + alpha_b, 1 <= b <= m, and the normalized monomial M_alpha(z) = sqrt((N+d-1)! / ((d-1)! alpha_1! ... alpha_d!)) z_1^alpha_1 ... z_d^alpha_d. Let F be the normalized Fourier matrix, with entries F_(t,s) = L^(-1/2) exp(2 pi i t s/L), for 1 <= t,s <= L. For each increasing m-element subset T =…
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