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Reproducing_kernel_Hilbert_spaces-print_book

reproducing

kernel,

reproducing

kernel

Hilbert

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In this thesis we make a survey of the theory of reproducting kernel hilbert spaces associated with positive definite kernels and we illustrate their applications for interpolation problems of nevanlinna-Pick type.

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REPRODUCING KERNEL HILBERT SPACES

a thesis

submitted to the department of mathematics

and the institute of engineering and science

of bilkent university

in partial fulfillment of the requirements

for the degree of

master of science

Baver Okutmu¸stur

August, 2005

I certify that I have read this thesis and that in my opinion it is fully adequate,

in scope and in quality, as a thesis for the degree of Master of Science.

Assist. Prof. Dr. Aurelian Gheondea (Supervisor)

I certify that I have read this thesis and that in my opinion it is fully adequate,

in scope and in quality, as a thesis for the degree of Master of Science.

Prof. Dr. Mefharet Kocatepe

I certify that I have read this thesis and that in my opinion it is fully adequate,

in scope and in quality, as a thesis for the degree of Master of Science.

Assoc. Prof. Dr. H. Turgay Kaptano˘glu

Approved for the Institute of Engineering and Science:

Prof. Dr. Mehmet B. Baray

Director of the Institute Engineering and Science

ABSTRACT

REPRODUCING KERNEL HILBERT SPACES

Baver Okutmu¸stur

M.S. in Mathematics

Supervisor: Assist. Prof. Dr. Aurelian Gheondea

August, 2005

In this thesis we make a survey of the theory of reproducing kernel Hilbert spaces

associated with positive deﬁnite kernels and we illustrate their applications for in-

terpolation problems of Nevanlinna-Pick type. Firstly we focus on the properties

of reproducing kernel Hilbert spaces, generation of new spaces and relationships

between their kernels and some theorems on extensions of functions and kernels.

One of the most useful reproducing kernel Hilbert spaces, the Bergman space, is

studied in details in chapter 3. After giving a brief deﬁnition of Hardy spaces, we

dedicate the last part for applications of interpolation problems of Nevanlinna-

Pick type with three main theorems: interpolation with a ﬁnite number of points,

interpolation with an inﬁnite number of points and interpolation with points on

the boundary. Finally we include an Appendix that contains a brief recall of the

main results from functional analysis and operator theory.

Keywords: Reproducing kernel, Reproducing kernel Hilbert spaces, Bergman

spaces, Hardy spaces, Interpolation, Riesz theorem.

iii

OZET

GURAN C¸ EK

IRDEKL

I H

ILBERT UZAYLARI

Baver Okutmu¸stur

Matematik, Y¨uksek Lisans

Tez Y¨oneticisi: Yrd. Do¸c. Dr. Aurelian Gheondea

A˘gustos, 2005

Bu tezde, do˘guran ¸cekirdekli Hilbert uzayları teorisini pozitif tanımlı

¸cekirdekler ile beraber inceledik ve bunun uygulamalarını Nevallina-Pick inter-

polasyon problemleri ¨uzerinde ¨ornekledik.

Oncelikle, do˘guran ¸cekirdekli Hilbert

uzaylarının ¨ozelliklerini, ¨uretilen yeni uzaylar ve onların ¸cekirdekleri arasındaki

ili¸skileri ve geni¸sletilen ¸ce¸sitli fonksiyon ve ¸cekirdeklerle ilgili bazı teoremleri

inceledik. Sık¸ca kullanılan do˘guran ¸cekirdekli Hilbert uzaylarından biri olan

Bergman uzayı 3. kısımda detaylarıyla i¸slendi. Hardy uzayının kısa bir

tanımıyla ba¸sladı˘gımız son kısım, Nevallina-Pick interpolasyon problemlerinin

uygulamalarını i¸ceren ¨u¸c ana teorem ile son buldu. Bunlar: sınırlı sayıda nokta

ile interpolasyon, sınırsız sayıda nokta ile interpolasyon ve sınır noktalarında in-

terpolasyon. Son olarak Appendix kısmı bu tezde sık¸ca kullandı˜gımız fonksiyonel

analiz ve operator teori ile ilgili temel esasların kısa bir ¨ozetine ayrıldı.

Anahtar s¨ozc¨ukler: Do˘guran ¸cekirdekler, Do˘guran ¸cekirdekli Hilbert uzayları,

Bergman uzayları, Hardy uzayları,

Interpolasyon, Riesz teoremi.

Acknowledgement

I would like to express my sincere gratitude to my supervisor Assoc. Prof.

Aurelian Gheondea, ﬁrstly for introducing me to this area of analysis, for his

excellent guidance, valuable suggestions, encouragement and inﬁnite patience. I

am glad to have a chance to study with this great person who is a role model as

a supervisor and a mathematician.

My sincere thanks to Prof. Dr. Mefharet Kocatepe for her interest and useful

comments.

I also need to express my gratitude to Assoc. Prof. H. Turgay Kaptano˜glu

for providing me necessary documents for this study.

I am grateful to Aslı Pekcan and Murat Altunbulak who helped me about

Latex. Special thanks to Aslı for reading my thesis, for her advices, corrections

and helps.

Beside, I would like to thank Serdar, Hikmet, Rohat, Erg¨un,

Inan, Fatma,

Burcu, Sultan and all my friends who always cared about my work and increased

my motivation, which I have strongly needed.

I would like to thank Eliya B¨uy¨ukkaya who has always been with me with her

endless love, support and understanding.

Finally, I would like to express my deep est gratitude for the constant support,

encouragement, trust and love that I received from my family.

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