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Approximation, interpolation, and lifting on the unit ball

Year: 2025

Type: preprint

Abstract: 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 Carathéodory 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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2512.11349

Authors: Bhattacharjee, Jaydeep, D., Deepak K., Sarkar, Jaydeb

Institutions: National Yang Ming Chiao Tung University, Indian Statistical Institute

Topic: Holomorphic and Operator Theory

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Arthritis National Research Foundation, Department of Science and Technology, Ministry of Science and Technology, India

OpenAlex ID: W7115599286

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