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AuthorLelong, Pierre. author
TitleEntire Functions of Several Complex Variables [electronic resource] / by Pierre Lelong, Lawrence Gruman
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1986
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Descript XII, 272 p. online resource


I - Entire functions of several complex variables constitute an important and original chapter in complex analysis. The study is often motivated by certain applications to specific problems in other areas of mathematics: partial differential equations via the Fourier-Laplace transformation and convolution operators, analytic number theory and problems of transcenยญ dence, or approximation theory, just to name a few. What is important for these applications is to find solutions which satisfy certain growth conditions. The specific problem defines inherently a growth scale, and one seeks a solution of the problem which satisfies certain growth conditions on this scale, and sometimes solutions of minimal asympยญ totic growth or optimal solutions in some sense. For one complex variable the study of solutions with growth conditions forms the core of the classical theory of entire functions and, historically, the relationship between the number of zeros of an entire function f(z) of one complex variable and the growth of If I (or equivalently log If I) was the first example of a systematic study of growth conditions in a general setting. Problems with growth conditions on the solutions demand much more precise information than existence theorems. The correspondence between two scales of growth can be interpreted often as a correspondence between families of bounded sets in certain Frechet spaces. However, for applications it is of utmost importance to develop precise and explicit representations of the solutions


1. Measures of Growth -- ยง1. Preliminaries -- ยง 2. Subharmonic and Plurisubharmonic Functions -- ยง3. Norms on ?n and Order of Growth -- ยง4. Minimal Growth: Liouvilleโ{128}{153}s Theorem and Generalizations -- ยง 5. Entire Functions of Finite Order -- ยง6. Proximate Orders -- ยง7. Regularizations -- ยง 8. Indicator of Growth Functions -- ยง 9. Exceptional Sets for Growth Conditions -- Historical Notes -- 2. Local Metric Properties of Zero Sets and Positive Closed Currents -- ยง1. Positive Currents -- ยง2. Exterior Product -- ยง3. Positive Closed Currents -- ยง 4. Positive Closed Currents of Degree 1 -- ยง5. Analytic Varieties and Currents of Integration -- Historical Notes -- 3. The Relationship Between the Growth of an Entire Function and the Growth of its Zero Set -- ยง1. Positive Closed Currents of Degree 1 Associated with a Positive Divisor -- ยง 2. Indicators of Growth of Cousin Data in ?n -- ยง3. Canonical Potentials in ?m -- ยง 4. The Canonical Representation of Entire Functions of Finite Order -- ยง5. Solution of the ? $$\bar \partial$$ Equation -- ยง 6. The Case of a Cousin Data -- ยง7. Slowly Increasing Cousin Data: the Genus q = 0; the Algebraic Case -- ยง8. The Case of Integral Order: Extension of a Theorem of Lindelรถf -- ยง 9. Trace of a Cousin Data on Complex Lines -- ยง 10. The Case of a Cousin Data of Infinite Order -- Historical Notes -- 4. Functions of Regular Growth -- ยง 1. General Properties of Functions of Regular growth -- ยง2. Distribution of the Zeros of Functions of Regular Growth -- Historical Notes -- 5. Holomorphic Mappings from ?n to ?m -- ยง1. Representation of an Analytic Variety Y in ?n as F-1(0) -- ยง2. Local Potentials and the Defect of Plurisubharmonicity -- ยง3. Global Potentials -- ยง 4. Construction of a System F of Entire Functions such that Y=F-1(0) -- ยง5. The Case of Slow Growth -- ยง6. The Algebraic Case -- ยง7. The Pseudo Algebraic Case -- ยง8. Counterexamples to Uniform Upper Bounds -- ยง9. An Upper Bound for the Area of F-1(a) for a Holomorphic Map -- ยง 10. Upper and Lower Bounds for the Trace of an Analytic Variety on Complex Planes -- Historical Notes -- 6. Application of Entire Functions in Number Theory -- ยง1. Preliminaries from Number Theory -- ยง2. A Schwarz Lemma -- ยง3. Statement and Proof of the Main Theorem -- Historical Notes -- 7. The Indicator of Growth Theorem -- Historical Notes -- 8. Analytic Functionals -- ยง1. Convex Sets and the Fourier-Borel Transform -- ยง2. The Projective Indicator -- ยง3. The Projective Laplace Transform -- ยง4. The Case of M a Complex Submanifold of ?n -- ยง5. The Generalized Laplace Transform and Indicator Function -- ยง6. Support for Analytic Functionals -- ยง7. Unique Supports for Domains in ?n -- ยง8. Unique Convex Supports -- Historical Notes -- 9. Convolution Operators on Linear Spaces of Entire Functions -- ยง1. Linear Topological Spaces of Entire Functions -- ยง 2. Theorems of Division -- ยง3. Applications of Convolution Operators in the Spaces Ep?(r)and Eo -- ยง4. Supplementary Results for Proximate Orders with ?>1 -- ยง5. The Case ?=1 -- ยง 6. More on Functions of Order Less than One -- ยง 7. Convolution Operators in ?n -- Historical Notes -- Appendix I. Subharmonic and Plurisubharmonic Functions -- Appendix II. The Existence of Proximate Orders

Mathematics Functions of complex variables Mathematics Several Complex Variables and Analytic Spaces


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