Optimal numerical approximation of bounded linear functionals by weighted sums in Hilbert spaces of functions defined in a domain B ⊂ C or B ⊂ Rm, invariant in rotation or translation (e.g. circle, circular annulus, b...
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Let Omega subset of C-m be a bounded connected open set and H subset of O(Omega) be an analytic Hilbert module, i.e., the Hilbert space possesses a reproducingkernel K, the polynomial ring C[z] subset of H is dense a...
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Let Omega subset of C-m be a bounded connected open set and H subset of O(Omega) be an analytic Hilbert module, i.e., the Hilbert space possesses a reproducingkernel K, the polynomial ring C[z] subset of H is dense and the point-wise multiplication induced by p is an element of C[z] is bounded on H. We fix an ideal I subset of C[z] generated by p(1), ...,p(t) and let [I] denote the completion of I in H. The sheaf S-H associated to analytic Hilbert module H is the sheaf O(Omega) of holomorphic functions on Omega and hence is free. However, the subsheaf S-[I] associated to [I] is coherent and not necessarily locally free. Building on the earlier work of Biswas, Misra and Putinar (Journal fr die reine and angewandte Mathematik (Crelles Journal) 662:165-204, 2012), we prescribe a hermitian structure for a coherent sheaf and use it to find tractable invariants. Moreover, we prove that if the zero set V-[I] is a submanifold of codimension t, then there is a unique local decomposition for the kernel K-[I(]) along the zero set that serves as a holomorphic frame for a vector bundle on V-[I]. The complex geometric invariants of this vector bundle are also unitary invariants for the submodule [I] subset of H.
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