arXiv subjects

Mahishanka Withanachchi

Publications and source records attributed to Mahishanka Withanachchi.

3 recordsLinked to original sources

arXiv2025

Solvability of the B\'ezout Equation for Banach Algebra-Valued $H^\infty$ Functions on the Polydisk

In connection with the still unsolved multidimensional corona problem for algebras of bounded holomorphic functions on convex domains, we study the solvability of the B\'ezout equation for the algebra of bounded holomorphic functions on the polydisk with values in a complex Banach algebra. Assuming local solvability of the B\'ezout equation on a special open cover of the maximal ideal space of the algebra, we combine a dimension-induction scheme with a careful analysis of the topological structure of this space to glue local solutions into a global one. As a corollary, we obtain the solvability of the B\'ezout equation for a broader class of subalgebras containing the slice algebra of bounded holomorphic functions, the case of the latter having been previously proved by the first author

math.CV↗

arXiv2025

Learning Spatio-Temporal Dynamics via Operator-Valued RKHS and Kernel Koopman Methods

We introduce a unified framework for learning the spatio-temporal dynamics of vector valued functions by combining operator valued reproducing kernel Hilbert spaces (OV-RKHS) with kernel based Koopman operator methods. The approach enables nonparametric and data driven estimation of complex time evolving vector fields while preserving both spatial and temporal structure. We establish representer theorems for time dependent OV-RKHS interpolation, derive Sobolev type approximation bounds for smooth vector fields, and provide spectral convergence guarantees for kernel Koopman operator approximations. This framework supports efficient reduced order modeling and long term prediction of high dimensional nonlinear systems, offering theoretically grounded tools for forecasting, control, and uncertainty quantification in spatio-temporal machine learning.

cs.LG↗

arXiv2025

Spatio-Temporal Prediction via Operator-Valued RKHS and Koopman Approximation

We develop a comprehensive framework for spatio-temporal prediction of time-varying vector fields using operator-valued reproducing kernel Hilbert spaces (OV RKHS). By integrating Sobolev regularity with Koopman operator theory, we establish representer theorems, approximation rates, and spectral convergence results for kernel-based learning of dynamical systems. Our theoretical contributions include new representer theorems for time-aligned OV RKHS interpolation, Sobolev approximation bounds for smooth vector fields, kernel Koopman operator approximations, and spectral convergence guarantees. These results underpin data-driven reduced-order modeling and forecasting for complex nonlinear dynamical systems.

math.GM↗