4 research outputs found
A new realization of rational functions, with applications to linear combination interpolation
We introduce the following linear combination interpolation problem (LCI):
Given distinct numbers and complex numbers
and , find all functions analytic in a simply
connected set (depending on ) containing the points such
that To this end we prove a representation
theorem for such functions in terms of an associated polynomial . We
first introduce the following two operations, substitution of , and
multiplication by monomials . Then let be the
module generated by these two operations, acting on functions analytic near
. We prove that every function , analytic in a neighborhood of the roots
of , is in . In fact, this representation of is unique. To solve the
above interpolation problem, we employ an adapted systems theoretic
realization, as well as an associated representation of the Cuntz relations
(from multi-variable operator theory.) We study these operations in reproducing
kernel Hilbert space): We give necessary and sufficient condition for existence
of realizations of these representation of the Cuntz relations by operators in
certain reproducing kernel Hilbert spaces, and offer infinite product
factorizations of the corresponding kernels
Coefficient-Based Regression with Non-Identical Unbounded Sampling
We investigate a coefficient-based least squares regression problem with indefinite kernels from non-identical unbounded sampling processes. Here non-identical unbounded sampling means the
samples are drawn independently but not identically from unbounded sampling processes. The kernel is not necessarily symmetric or positive semi-definite. This leads to additional difficulty in the error analysis. By introducing a suitable reproducing kernel Hilbert space (RKHS) and a suitable intermediate integral operator, elaborate analysis is presented by means of a novel technique for the sample error. This leads to satisfactory results