## Abstract

In a seminal paper, Sarason generalized some classical interpolation problems for H^{∞} functions on the unit disc to problems concerning lifting onto H^{2} of an operator T that is defined on K = H^{2}⊖φH^{2} (φ is an inner function) and commutes with the (compressed) shift 5. In particular, he showed that interpolants (i.e., f ∈ H^{∞} such that f(S) = T) having norm equal to ∥T∥ exist, and that in certain cases such an / is unique and can be expressed as a fraction f = b/a with a, 6 ∈ K. In this paper, we study interpolants that are such fractions of K functions and are bounded in norm by 1 (assuming that ∥T∥ < 1, in which case they always exist). We parameterize the collection of all such pairs (a, b) ∈ K × K and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where 0 is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.

Original language | English (US) |
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Pages (from-to) | 965-987 |

Number of pages | 23 |

Journal | Transactions of the American Mathematical Society |

Volume | 358 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2006 |

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