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↗