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A companion volume to the text "Complex Variables: An Introduction" by the same authors, this book further develops the theory, continuing to emphasize the role that the Cauchy-Riemann equation plays in modern complex analysis. Topics considered include: Boundary values of holomorphic functions in the sense of distributions; interpolation problems and ideal theory in algebras of entire functions with growth conditions; exponential polynomials; the G transform and the unifying role it plays in complex analysis and transcendental number theory; summation methods; and the theorem of L. Schwarz concerning the solutions of a homogeneous convolution equation on the real line and its applications in harmonic function theory.



Chapter 1. Boundary Values of Holomorphic Functions and Analytic Functionals

The Schwarz Reflection Principle leads naturally to the consideration of boundary values of holomorphic functions. Those boundary values can exist pointwise, almost everywhere, or in some generalized sense, for instance, in the sense of distributions, as in the Edge-of-the-Wedge Theorem (see [BG, Theorem 3.6.23], [Beur]). Let us make these concepts more precise.
Carlos A. Berenstein, Roger Gay

Chapter 2. Interpolation and the Algebras AP

In the first chapter, we have seen how the Leitmotiv of the boundary values of holomorphic functions lead us naturally to introduce several transforms, in particular, the Fourier-Borel and Fourier transforms, and found out that many questions can be posed in equivalent terms in the algebras of entire functions with growth conditions, Exp(Ω) and F(ɛ’(ℝ)), specially problems relating to convolution equations. In the case of distributions, this relation will be come more evident in Chapter 6. The aim of this chapter is to study a more general class of algebras, the Hörmander algebras, A p (Ω). We shall see that the ideal theory of these algebras is intimately related to the study of interpolation varieties. In the previous volume [BG, Chapter 3], we have shown that to be the case for the algebras of holomorphic functions (Ω), and we found out that one could study interpolation questions with the help of the inhomogeneous Cauchy-Riemann equation. The same will be the case here. This time, though, we shall be obliged to consider the problem of solving the Cauchy-Riemann equation with growth constraints.
Carlos A. Berenstein, Roger Gay

Chapter 3. Exponential Polynomials

An exponential polynomial is an entire function f of the form
$$ f(z) = \sum\limits_{{1 \leqslant j \leqslant m}} {{{P}_{j}}} (z){{e}^{{ajz}}}, $$
where α j ∈ ℂ, P j ∈ ℂ[z]. We assume that the α j are distinct and the polynomials P j not zero. The P j are called the coefficients of f and α j the frequencies. (Sometimes the α j are called the exponents especially in the Russian literature. In some contexts α j = iλ j j ∈ℝ , and the λ j are called the frequencies and τ j = 2π j (when λ j ≠ 0) the periods; clearly eiλjz periodic of period τ j .) It is immediate that there is a unique analytic functional T whose Fourier—Borel transform F (T) coincides with f, i.e.,F(T) (z) = 〈T ζ , e〉 = f (z). Namely, T is representable as a distribution in ℂ of the form
$$T = \sum\limits_{j,v} {{a_{j,v}}\delta _{\alpha j}^{(v)}} $$
where δα j is the Dirac measure at the point α j , δ αj (υ) is a “holomorphic” derivative of order υ, δ αj (υ) = (∂/∂z) υ δα j , and the aj,υ are complex constants. In the case where the frequencies α j are purely imaginary, i.e., α j = iλj, λ j ∈ ℝthen f is the Fourier transform of a distribution μέ(ℝ). That is, we let
$$\mu :\sum\limits_{j,v} {{a_{j,v}}{i^v}\frac{{{d^v}}}{{d{x^v}}}{\delta _{\lambda j}}} $$
with δλj the Dirac mass at the point λ j ∈ℝ (acting on C∞ functions in ℝ and
$$f(z) = < {\mu _x},{e^{ - ixz}} >$$
Carlos A. Berenstein, Roger Gay

Chapter 4. Integral Valued Entire Functions

In this section we study a transform of analytic functionals akin to the Cauchy transform considered in Chapter 1. This transform will allow us to obtain rather easily those properties of entire functions of exponential type that can be derived from their behavior on sequences of the form nn0, n ∈ ℤ. It also provides an elementary method to study the analytic continuation of power series of the form Σ≥0 f (n)t n , where f is an entire function of exponential type. The main references for this section are [Bo], [Av 1], [Av2], [AG1], [AG2], [AG3].
Carlos A. Berenstein, Roger Gay

Chapter 5. Summation Methods

Given a power series \( f\left( z \right) = \sum\nolimits_{n \geqslant 0} {{a_n}} {z^n}\) of radius of convergence R, 0 < R < ∞, we are trying to find explicitly the analytic continuation of f to the largest domain, star-shaped with respect to the origin, to which f admits an analytic continuation. Let us denote by D(f) that domain. (Why is it well defined?) We shall obtain D(f) as the union of certain domains B ρ (f),such that in each of them we shall be able to describe explicitly the analytic continuation of f, these domains are parametrized by ρ ≥ 1. The domain D(f) is called the star of holomorphy of f. We start by explaining how to determine B(f) = B1 (f),usually called the Borel polygon of f.
Carlos A. Berenstein, Roger Gay

Chapter 6. Harmonic Analysis

The origins of Harmonic Analysis lie in the work of Euler [Eu] and the Bernoullis who proposed to write periodic functions in terms of the exponentials e inx , n ∈ ℤ, in their study of the vibrating string. It is known that every C-function which is 2π-periodic in the real line has an expansion of the form En \(\sum\nolimits_{n = - \infty }^{ + \infty } {{a_n}{e^{inx}}} \) (we remind the reader one can estimate these coefficients a n very precisely, and that we do not need to restrict ourselves to C-functions). It was the work of Fourier [Fo] on heat conduction that showed, once and for all, the importance and the interest of such expansions, and since then they have been called Fourier expansions. It is clear that another way of saying that a function f is periodic with period τ is to say that f satisfies the convolution equation
$$\left( {{\delta _\tau } - \delta } \right) * f = 0$$
Carlos A. Berenstein, Roger Gay


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