Complex Analysis and Special Topics in Harmonic Analysis

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.

Inhalt
1 Boundary Values of Holomorphic Functions and Analytic Functionals.- 1.1. The Hardy Spaces in the Disk.- 1.2. Hyperfunctions.- 1.3. Analytic Functionals and Entire Functions of Exponential Type.- 1.4. Vade Mecum of Functional Analysis.- 1.5. Convolution of Analytic Functionals.- 1.6. Analytic Functionals on the Unit Circle.- 2 Interpolation and the Algebras Ap.- 2.1. The Algebras Ap.- 2.2. Interpolation with Growth Conditions.- 2.3. Ideal Theory in Ap.- 2.4. Dense Ideals in Ap(?).- 2.5. Local Ideals and Conductor Ideals in Ap.- 2.6. The Algebra A? of Entire Functions of Order at Most ?.- 3 Exponential Polynomials.- 3.1. The Ring of Exponential Polynomials.- 3.2. Distributions of Zeros of an Exponential Polynomial.- 4 Integral Valued Entire Functions.- 4.1. The G-Transform.- 4.2. Integral Valued Entire Functions.- 5 Summation Methods.- 5.1. Borel and MittagLeffler Summation Methods.- 5.2. The Lindelöf Indicator Function.- 5.3. The FourierBorel Transform of Order ? of Analytic Functionals.- 5.4. Analytic Functionals with Noncompact Carrier.- 6 Harmonic Analysis.- 6.1. Convolution Equations in ?.- 6.2. Convolution Equations in ?.- 6.3. The Equation f(z + 1) f(z) = g(z).- 6.4. Differential Operators of Infinite Order.- 6.5. Deconvolution.- References.- Notation.