This revised and enlarged second edition is devoted to asymptotical questions of the theory of entire and plurisubharmonic functions. A separate chapter deals with applications in biophysics. The book is of interest to research specialists in theoretical and applied mathematics, postgraduates and students who are interested in complex and real analysis and its applications.
Table of ContentsFrom the Editors of the Russian Edition. Foreword to the Russian Edition. Introduction.
1: Preliminary. 1.1. On the growth of nondecreasing functions of one variable. 1.2. Semicontinuous functions. 1.3. Convex sets and associated functions. 1.4. Convex functions. 1.5. Duality of convex functions. 1.6. Asymptotic properties of convex functions. 1.7. Minkowski theorem on convex bodies. 1.8. Plurisubharmonic functions. 1.9. Trigonometrically rho-convex functions. 1.10. Selected facts about the entire functions of one variable.
2: A Method of Identifying Homeostasis Relaxation Characteristics. 2.1. Homeostasis system relaxation characteristics and the problem of their identification. 2.2. Algorithm of recovering a quasipolynomial by its moments. 2.3. Algorithm of approximation of discrete functions by quasipolynomials (identification algorithm). 2.4. The case of quasipolynomials of order 2 and 3. 2.5. The case of warp-type homeostasis processes.
3: Indicator Diagram of an Entire Function of One Variable with Nonnegative indicator. 3.1. Plane rho-convex sets and the indicator diagram. 3.2. Analog of the Polya theorem for an entire function of order rho NOT= 1 and with nonnegative indicator. 3.3. The generalized Borel polygon of a power series.
4: Plane Indicator Diagram of Entire Function of Order rho > 0 with the Indicator of General Form. 4.1. Minimal trigonometrically rho-convex functions. 4.2. Many-sheeted diagrams associated with the functions of class Ppi. 4.3. The relationship of the polynomials alpha(z) = zro-1 +...+ anzro-n. 4.4. Plane (ro,alpha)-convex sets and the plane indicator diagram of an entire function of order ro greater than 0.
5: Spaces of Entire Functions of Order ro greater than 0 with Restrictions on the Indicator. 5.1. Entire function of two variables associated with a polynomial in ro(l). 5.2. The analog of Borel's transformation and realization of the spaces [ro, h(theta)], [ro, h(theta)]. 5.3. Applications of the analog of the Polya theorem.
6: Geometrical Analysis of Asymptotics of Functions Plurisubharmonic in C;n. 6.1. Simplest properties of functions of classes B, U. 6.2. Various definitions of orders of functions of class U. 6.3. Local APHi-type structure for function PHi in U. 6.4. Global APHi-type structure for function PHi in U.
7: Growth Characteristics of Entire Functions (Orders, Types) and Their Applications. 7.1. Relationship between the growth characteristics of an entire function and its Taylor coefficients. 7.2. Existence of entire functions with prescribed growth characteristics. 7.3. The modulus maximum and the maximum term of an entire function: Comparative growth. 7.4. On the growth of the Nevanlinna characteristic for an entire function of several variables.
8: Indicator Diagram of an Entire Function of Several Variables with Nonnegative Indicator. 8.1. System of indicators and indicator diagrams of an entire function of several variables. 8.2. Circular sets and their properties.