arXiv subjects

Deepak K. D.

Publications and source records attributed to Deepak K. D..

3 recordsLinked to original sources

arXiv2025

Approximation, interpolation, and lifting on the unit ball

We solve the Nevanlinna-Pick interpolation problem on the open unit ball of the complex $n$-space. Our solutions signify the role of inner functions on the unit ball, objects whose existence was once considered uncertain. The results also reveal the importance of extremal functions, which emerge as natural analogues of finite Blaschke products. This viewpoint is illustrated by the Carath\'{e}odory approximation and Pick's theorems on the unit ball. We solve the commutant lifting problem, where both inner and extremal functions play a fundamental role. These results resolve several well-known problems on the unit ball.

math.CV↗

arXiv2023

Commutant lifting, interpolation, and perturbations on the polydisc

The fundamental theorem on commutant lifting due to Sarason does not carry over to the setting of the polydisc. This paper presents two classifications of commutant lifting in several variables. The first classification links the lifting problem to the contractivity of certain linear functionals. The second one transforms it into nonnegative real numbers via a distance formula. We also solve the Nevanlinna-Pick interpolation problem for bounded analytic functions on the polydisc. Along the way, we solve a perturbation problem for bounded analytic functions. Commutant lifting and interpolation on the polydisc solve two well-known problems in Hilbert function space theory.

math.FA↗

arXiv2019

Commutant lifting and Nevanlinna-Pick interpolation in several variables

This paper concerns a commutant lifting theorem and a Nevanlinna-Pick type interpolation result in the setting of multipliers from vector-valued Drury-Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in $\mathbb{C}^n$. The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball.

math.FA↗