{"id":"palomar/PALOMAR-2026-10-09-000004","url":"https://github.com/mrklwk/uniformly-bounded-holomorphic-bases/tree/a91b7a0f08c5771f1c66435eac4a456a918ff384","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-09-000004-v2.json","repo":"https://github.com/mrklwk/uniformly-bounded-holomorphic-bases","tags":["math.CV","math.FA","complex-analysis","holomorphic-polynomials","orthonormal-bases"],"score":0,"title":"Explicit uniformly bounded bases of homogeneous holomorphic polynomials in every dimension","trust":"high","domain":"math.CV","source":"palomar","authors":["Mark Lewko"],"reviews":0,"subject":"math.CV","summary":"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 =…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-11","reactions":{},"formalized":true,"formalizes":[{"title":"Nonpublic mathematical source","identifier":"","relationship":"formalizes"}]}