Entire Functions of Several Complex Variables

1. Front Matter

   Pages I-XI

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2. Measures of Growth

   • Pierre Lelong, Lawrence Gruman

   Pages 1-29

3. Local Metric Properties of Zero Sets and Positive Closed Currents

   • Pierre Lelong, Lawrence Gruman

   Pages 30-58

4. The Relationship Between the Growth of an Entire Function and the Growth of its Zero Set

   • Pierre Lelong, Lawrence Gruman

   Pages 59-94

5. Functions of Regular Growth

   • Pierre Lelong, Lawrence Gruman

   Pages 95-115

6. Holomorphic Mappings from ℂⁿ to ℂ^m

   • Pierre Lelong, Lawrence Gruman

   Pages 116-154

7. Application of Entire Functions in Number Theory

   • Pierre Lelong, Lawrence Gruman

   Pages 155-166

8. The Indicator of Growth Theorem

   • Pierre Lelong, Lawrence Gruman

   Pages 167-176

9. Analytic Functionals

   • Pierre Lelong, Lawrence Gruman

   Pages 177-200

10. Convolution Operators on Linear Spaces of Entire Functions

    • Pierre Lelong, Lawrence Gruman

    Pages 201-229

11. Back Matter

    Pages 230-272

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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 transcen­ 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 asymp­ 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.

• Area
• Complex analysis
• Frechet Spaces
• Functions
• Lelong
• Schwarz lemma
• Variables
• analytic number theory
• approximation
• boundary element method
• equation
• function
• information
• number theory
• presentation

• Université Paris VI, Paris Cedex 05, France

  Pierre Lelong

• UER de mathématiques, C.N.R.S., Université de Provence, Marseille, France

  Lawrence Gruman

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