Generalisations of Pick's theorem to reproducing Kernel Hilbert spaces

Mathematics and Statistics

Electronic data

Quiggin
Final published version, 528 KB, PDF document

Research output: Thesis › Doctoral Thesis

Published

Standard

Generalisations of Pick's theorem to reproducing Kernel Hilbert spaces. / Quiggin, Peter Philip.
Lancaster: Lancaster University, 1994. 120 p.

Research output: Thesis › Doctoral Thesis

Harvard

Quiggin, PP 1994, 'Generalisations of Pick's theorem to reproducing Kernel Hilbert spaces', PhD, Lancaster University, Lancaster.

APA

Quiggin, P. P. (1994). Generalisations of Pick's theorem to reproducing Kernel Hilbert spaces. [Doctoral Thesis, Lancaster University]. Lancaster University.

Vancouver

Quiggin PP. Generalisations of Pick's theorem to reproducing Kernel Hilbert spaces. Lancaster: Lancaster University, 1994. 120 p.

Author

Quiggin, Peter Philip. / Generalisations of Pick's theorem to reproducing Kernel Hilbert spaces. Lancaster : Lancaster University, 1994. 120 p.

Bibtex

@phdthesis{76b15f9422ec47a1b22ae989a3062bb0,

title = "Generalisations of Pick's theorem to reproducing Kernel Hilbert spaces",

abstract = "Pick's theorem states that there exists a function in H1, which is bounded by 1and 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 generalisationwhen H1 is replaced by the space of multipliers of a general reproducing kernelHilbert space H(K) (where K is the reproducing kernel). J. Agler showed that thisgeneralised theorem is true when H(K) is a certain Sobolev space or the Dirichletspace. This thesis widens Agler's approach to cover reproducing kernel Hilbertspaces 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 arethen used to prove Pick's theorem for certain weighted Hardy and Sobolev spacesand for a functional Hilbert space introduced by Saitoh. The reproducing kernelapproach is then used to derived results for several related problems. These includethe uniqueness of the optimal interpolating multiplier, the case of operator-valuedfunctions and a proof of the Adamyan-Arov-Kren theorem.",

author = "Quiggin, {Peter Philip}",

year = "1994",

language = "English",

publisher = "Lancaster University",

school = "Lancaster University",

}

RIS

TY - BOOK

T1 - Generalisations of Pick's theorem to reproducing Kernel Hilbert spaces

AU - Quiggin, Peter Philip

PY - 1994

Y1 - 1994

N2 - Pick's theorem states that there exists a function in H1, which is bounded by 1and 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 generalisationwhen H1 is replaced by the space of multipliers of a general reproducing kernelHilbert space H(K) (where K is the reproducing kernel). J. Agler showed that thisgeneralised theorem is true when H(K) is a certain Sobolev space or the Dirichletspace. This thesis widens Agler's approach to cover reproducing kernel Hilbertspaces 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 arethen used to prove Pick's theorem for certain weighted Hardy and Sobolev spacesand for a functional Hilbert space introduced by Saitoh. The reproducing kernelapproach is then used to derived results for several related problems. These includethe uniqueness of the optimal interpolating multiplier, the case of operator-valuedfunctions and a proof of the Adamyan-Arov-Kren theorem.

AB - Pick's theorem states that there exists a function in H1, which is bounded by 1and 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 generalisationwhen H1 is replaced by the space of multipliers of a general reproducing kernelHilbert space H(K) (where K is the reproducing kernel). J. Agler showed that thisgeneralised theorem is true when H(K) is a certain Sobolev space or the Dirichletspace. This thesis widens Agler's approach to cover reproducing kernel Hilbertspaces 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 arethen used to prove Pick's theorem for certain weighted Hardy and Sobolev spacesand for a functional Hilbert space introduced by Saitoh. The reproducing kernelapproach is then used to derived results for several related problems. These includethe uniqueness of the optimal interpolating multiplier, the case of operator-valuedfunctions and a proof of the Adamyan-Arov-Kren theorem.

M3 - Doctoral Thesis

PB - Lancaster University

CY - Lancaster

ER -

Research

Electronic data