Numerical solutions of a boundary value problem on the sphere using radial basis functions

Boundary value problems on the unit sphere arise naturally in geophysics and oceanography when scientists model a physical quantity on large scales. Robust numerical methods play an important role in solving these problems. In this article, we construct numerical solutions to a boundary value problem defined on a spherical sub-domain (with a sufficiently smooth boundary) using radial basis functions (RBF). The error analysis between the exact solution and the approximation is provided. Numerical experiments are presented to confirm theoretical estimates

Spectral synthesis in de Branges spaces

We solve completely the spectral synthesis problem for reproducing kernels in the de Branges spaces $\mathcal{H}(E)$. Namely, we describe the de Branges spaces $\mathcal{H}(E)$ such that all $M$-bases of reproducing kernels (i.e., complete and minimal systems $\{k_\lambda\}_{\lambda\in\Lambda}$ with complete biorthogonal $\{g_\lambda\}_{\lambda\in\Lambda}$) are strong $M$-bases (i.e., every mixed system $\{k_\lambda\}_{\lambda\in\Lambda\setminus\tilde \Lambda} \cup\{g_\lambda\}_{\lambda\in \tilde \Lambda}$ is also complete). Surprisingly this property takes place only for two essentially different classes of de Branges spaces: spaces with finite spectral measure and spaces which are isomorphic to Fock-type spaces of entire functions. The first class goes back to de Branges himself, the second class appeared in a recent work of A. Borichev and Yu. Lyubarskii. Moreover, we are able to give a complete characterisation of this second class in terms of the spectral data for $\mathcal{H}(E)$. In addition, we obtain some results about possible codimension of mixed systems for a fixed de Branges space $\mathcal{H}(E)$, and prove that any minimal system of reproducing kernels in $\mathcal{H}(E)$ is contained in an exact system of reproducing kernels.Comment: 38 pages. Shortened text with streamlined proofs. This version is accepted for publication in "Geometric and Functional Analysis