Analytic Kramer sampling and quasi Lagrange-type interpolation in vector valued RKHS

TL;DR

This paper develops advanced sampling and interpolation methods in vector valued RKHS, connecting them with operator theory and de Branges spaces, enhancing understanding of function reconstruction in these spaces.

Contribution

It introduces an abstract Kramer sampling theorem and extends quasi Lagrange-type interpolation to vector valued entire functions within RKHS, linking these concepts with operator invariance and de Branges spaces.

Findings

Established an abstract Kramer sampling theorem for vector valued RKHS.

Extended quasi Lagrange-type interpolation to vector valued entire functions.

Connected interpolation methods with operator invariance and de Branges spaces.

Abstract

This paper discusses an abstract Kramer sampling theorem for functions within a reproducing kernel Hilbert space (RKHS) of vector valued holomorphic functions. Additionally, we extend the concept of quasi Lagrange-type interpolation for functions within a RKHS of vector valued entire functions. The dependence of having quasi Lagrange-type interpolation on an invariance condition under the generalized backward shift operator has also been discussed. Furthermore, the paper establishes the connection between quasi Lagrange-type interpolation, operator of multiplication by the independent variable, and de Branges spaces of vector valued entire functions.