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Advanced Complex Analysis Problem Book: Topological Vector Spaces, Functional Analysis, and Hilbert Spaces of Analytic Functions 1st ed. 2015 [Pehme köide]

  • Formaat: Paperback / softback, 521 pages, kõrgus x laius: 240x168 mm, kaal: 8806 g, IX, 521 p., 1 Paperback / softback
  • Ilmumisaeg: 19-Nov-2015
  • Kirjastus: Birkhauser Verlag AG
  • ISBN-10: 3319160583
  • ISBN-13: 9783319160580
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Advanced Complex Analysis Problem Book: Topological Vector Spaces, Functional Analysis, and Hilbert Spaces of Analytic Functions 1st ed. 2015Suurem pilt
  • Formaat: Paperback / softback, 521 pages, kõrgus x laius: 240x168 mm, kaal: 8806 g, IX, 521 p., 1 Paperback / softback
  • Ilmumisaeg: 19-Nov-2015
  • Kirjastus: Birkhauser Verlag AG
  • ISBN-10: 3319160583
  • ISBN-13: 9783319160580
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783319160580.html
Märksõnad:
This is an exercises book at the beginning graduate level, whose aim is to illustrate some of the connections between functional analysis and the theory of functions of one variable. A key role is played by the notions of positive definite kernels and of reproducing kernel Hilbert spaces. A number of facts from functional analysis and topological vector spaces are surveyed. Then, various Hilbert spaces of analytic functions are studied.

Prologue.- Part I, Analytic functions.- 1 Some algebra.- 2 Exercises in complex variables.- Part II, Topology and functional analysis.- 3 Topological spaces.- 4 Normed spaces.- 5 Locally convex topological vector spaces.- 6 Some functional analysis.- Part III, Hilbert spaces of analytic functions.- 7 Reproducing kernel Hilbert spaces.- 8 Hardy spaces.- 9 de Branges-Rovnyak spaces.- 10 Bergman spaces.- 11 Fock spaces.- Index.- Name index.- Notation index.

Arvustused

The aim of this book is to fill in this gap, i.e. to get students familiar with some notions of functional analysis in the context of spaces of analytic functions, based on the unifying idea of reproducing kernel Hilbert space. the book is dedicated to beginning graduate students aiming a specialization in complex analysis. Teachers of complex analysis will find some supplementary material here and those of functional analysis a source of concrete examples. (S. Cobza, Studia Universitatis Babes-Bolyai, Mathematica, Vol. 61 (1), 2016)

I Analytic Functions
9(110)
1 Algebraic Prerequisites
11(50)
1.1 Sets
11(4)
1.2 Groups
15(4)
1.3 Vector spaces
19(4)
1.4 Fields
23(1)
1.5 Matrices
24(4)
1.6 Positive matrices
28(6)
1.7 Rational functions
34(8)
1.8 Solutions of the exercises
42(19)
2 Analytic Functions
61(58)
2.1 Some warm-up exercises
61(14)
2.2 Some important theorems to recall
75(2)
2.3 Cauchy's formula and Laurent expansion
77(5)
2.4 Star-shaped and simply connected sets
82(2)
2.5 Various
84(1)
2.6 Harmonic and subharmonic functions
85(4)
2.7 Meromorphic functions in C
89(1)
2.8 Solutions of the exercises
90(29)
II Topology and Functional Analysis
119(210)
3 Topological Spaces
121(78)
3.1 Topological spaces
122(5)
3.2 Compact sets
127(4)
3.3 Connected sets
131(1)
3.4 Continuous maps
132(4)
3.5 Products
136(1)
3.6 Closed, open and proper maps
137(3)
3.7 Quotient topology
140(2)
3.8 Manifolds and surfaces
142(6)
3.9 Metric spaces
148(9)
3.10 Solutions of the exercises
157(42)
4 Normed Spaces
199(50)
4.1 Normed Banach and Hilbert spaces
199(8)
4.2 Operators on normed spaces
207(15)
4.3 Unbounded operators
222(3)
4.4 Indefinite inner product spaces
225(2)
4.5 Solutions of the exercises
227(22)
5 Locally Convex Topological Vector Spaces
249(36)
5.1 Topological vector spaces
249(4)
5.2 Countably normed spaces and Frechet spaces
253(4)
5.3 Operators in countably normed spaces
257(2)
5.4 Dual of a Frechet space
259(3)
5.5 Positive operators
262(2)
5.6 Topology of the space of analytic functions
264(3)
5.7 Normal families
267(1)
5.8 The dual of the space of analytic functions
268(1)
5.9 Solutions of the exercises
268(17)
6 Some Functional Analysis
285(44)
6.1 Fourier transform
285(13)
6.2 Stieltjes integral
298(4)
6.3 Density results
302(4)
6.4 Solutions of the exercises
306(23)
III Hilbert Spaces of Analytic Functions
329(158)
7 Reproducing Kernel Hilbert Spaces
331(74)
7.1 Positive definite kernels
332(6)
7.2 Examples of positive definite functions and kernels
338(9)
7.3 Conditionally negative functions
347(3)
7.4 Vector-valued functions
350(2)
7.5 Reproducing kernel Hilbert spaces
352(9)
7.6 Linear operators in reproducing kernel Hilbert spaces
361(6)
7.7 Finite-dimensional reproducing kernel spaces
367(9)
7.8 Solutions of the exercises
376(29)
8 Hardy Spaces
405(38)
8.1 Reproducing kernel Hilbert spaces of analytic functions
405(1)
8.2 The Hardy space of the open unit disk
406(3)
8.3 Some operator theory in H2(D)
409(3)
8.4 Composition operators
412(1)
8.5 Cuntz relations
413(2)
8.6 The Hardy space of the open upper half-plane
415(5)
8.7 The fractional Hardy space Hv2 of the open upper half-plane
420(1)
8.8 Solutions of the exercises
421(22)
9 De Branges--Rovnyak Spaces
443(18)
9.1 De Branges--Rovnyak spaces
443(5)
9.2 Interpolation
448(1)
9.3 Caratheodory's theorem
449(1)
9.4 A few words on Schur analysis
450(3)
9.5 Solutions of the exercises
453(8)
10 Bergman Spaces
461(14)
10.1 The Bergman space of analytic functions analytic in D
461(2)
10.2 The Bergman space of analytic functions analytic in an ellipse
463(1)
10.3 The Bergman space of the annulus
464(1)
10.4 The Bergman spaces of polyanalytic functions
464(2)
10.5 Solutions of the exercises
466(9)
11 Fock Spaces
475(12)
11.1 The Bargmann--Fock--Segal spaces of analytic functions
475(3)
11.2 The Bargmann--Fock--Segal spaces of polyanalytic functions
478(1)
11.3 Solutions of the exercises
479(8)
Bibliography 487(22)
Index 509(9)
Name Index 518(3)
Notation Index 521
Prof. Daniel Alpay is a faculty member of the department of mathematics at Ben Gurion University, Beer Sheva, Israel. He is the incumbent of the Earl Katz Family chair in algebraic system theory. He has a double formation of electrical engineer (Telecom Paris, graduated 1978) and mathematician (PhD, Weizmann Institute, 1986). His research includes operator theory, stochastic analysis, and the theory of linear systems. Daniel Alpay is one of the initiators and responsible of the dual track electrical-engineering mathematics at Ben-Gurion University.

He is the author of "A Complex Analysis Problem Book" (Birkhäuser, 2011). Together with co-authors, he has written four books and more than 220 research papers, and edited twelve books of research papers.