Home > Research > Publications & Outputs > Generalisations of Pick's theorem to reproducin...

Electronic data

  • Quiggin

    Final published version, 528 KB, PDF document

View graph of relations

Generalisations of Pick's theorem to reproducing Kernel Hilbert spaces

Research output: ThesisDoctoral Thesis

  • Peter Philip Quiggin
Publication date1994
Number of pages120
Awarding Institution
  • Young, Nicholas, Supervisor
Award date30/08/1994
Place of PublicationLancaster
  • Lancaster University
<mark>Original language</mark>English


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.