Complex Analysis in Several Variables

The material in this chapter intentionally interrupts our theoretical development about Hilbert spaces. There are at least two reasons for this interruption. One reason is that this material is a prerequisite for understanding the Hilbert space A2(Bn) of square-integrable holomorphic functions on th...

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Bibliographic Details
Published in:	Inequalities from Complex Analysis pp. 63 - 88
Main Author:	John P. D’Angelo
Format:	Book Chapter
Language:	English
Published:	Washington DC The Mathematical Association of America 2002 Mathematical Association of America
Edition:	1
Subjects:	Abstract spaces Addition Algebra Analytic functions Applied mathematics Applied statistics Arithmetic Banach space Complex analysis Complex variables Dimensional analysis Dimensionality Hilbert spaces MAA books Mathematical analysis Mathematical expressions Mathematical functions Mathematical objects Mathematical series Mathematical values Mathematical variables Mathematics Metric spaces Partial sums Polynomials Power series Pure mathematics Real and complex analysis Separable spaces Statistical physics Statistics Topological spaces linear functional holomorphic function multi-index notation generating function automorphism Hartogs, F linear fractional transformation mean-value property Poisson kernel complete orthonormal system beta function pseudoconvex Hilbert space open unit ball projection anti-holomorphic orthonormal system harmonic vector-valued reproducing property Cauchy-Schwarz inequality geometric series gamma function conjugate holomorphic finite type homogeneous Bergman kernel function partial sum domain region of convergence uniform convergence real part imaginary part Riesz representation
ISBN:	9780883850336, 0883850338
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Summary:	The material in this chapter intentionally interrupts our theoretical development about Hilbert spaces. There are at least two reasons for this interruption. One reason is that this material is a prerequisite for understanding the Hilbert space A2(Bn) of square-integrable holomorphic functions on the unit ball; this particular Hilbert space arises in a crucial way in the proof of Theorem VII.1.1, the main result in the book. A second reason is the feeling that abstract material doesn't firmly plant itself in one's mind unless it is augmented by concrete material. Conversely the presentation of concrete material benefits from appropriate abstract interludes. In this chapter we introduce holomorphic functions of several complex variables. This presentation provides only a brief introduction to the subject. Multi-index notation and issues involving calculus of several variables also appear here. Studying them allows us to provide a nice treatment of the gamma and beta functions. We use them to compute the Bergman kernel function for the unit ball, thereby reestablishing contact with Hilbert spaces. Holomorphic functions Our study of A2(Bn) requires us to first develop some basic information about holomorphic functions of several complex variables. As in one variable, holomorphic functions of several variables are locally represented by convergent power series. Although some formal aspects of the theories are the same, geometric considerations change considerably in the higher-dimensional theory. In order to avoid a profusion of indices we first introduce multiindex notation. This notation makes many computations in several variables both easier to perform and simpler to expose. Suppose z = (z1, ..., zn) ∈ Cn, and let (α1, ..., αn) be an n-tuple of nonnegative integers.
ISBN:	9780883850336 0883850338
DOI:	10.5948/UPO9780883859704.004