Use this URL to cite or link to this record in EThOS:
Title: Generalisations of Pick's theorem to reproducing Kernel Hilbert spaces
Author: Quiggin, Peter Philip
ISNI:       0000 0001 3503 6810
Awarding Body: Lancaster University
Current Institution: Lancaster University
Date of Award: 1994
Availability of Full Text:
Access from EThOS:
Access from Institution:
Pick's theorem states that there exists a function in H1, which is bounded by 1 and takes given values at given points, if and only if a certain matrix is positive. H1 is the space of multipliers of H2 and this theorem has a natural generalisation when H1 is replaced by the space of multipliers of a general reproducing kernel Hilbert space H(K) (where K is the reproducing kernel). J. Agler showed that this generalised theorem is true when H(K) is a certain Sobolev space or the Dirichlet space. This thesis widens Agler's approach to cover reproducing kernel Hilbert spaces in general and derives sucient (and usable) conditions on the kernel K, for the generalised Pick's theorem to be true for H(K). These conditions are then used to prove Pick's theorem for certain weighted Hardy and Sobolev spaces and for a functional Hilbert space introduced by Saitoh. The reproducing kernel approach is then used to derived results for several related problems. These include the uniqueness of the optimal interpolating multiplier, the case of operator-valued functions and a proof of the Adamyan-Arov-Kren theorem.
Supervisor: Young, Nicholas Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Pure mathematics