Riemann's Boundary Problem with Infinite Index

by Nikolaĭ Vasilʹevich Govorov

The book gives the first complete presentation of two closely connected subjects that are now used in various branches of complex analysis. In the first part the theory of analytic functions of completely regular growth in an angle is developed. Being a natural extension of the classical theory of entire functions of completely regular growth, this theory possesses principally new features due to the influence of boundary values on the sides of the angle. The second part contains the theory of the Riemann boundary value problem with power type infinite index. Using the results obtained in the first part of the book, the author gives complete and efficient solutions to the problem in various natural classes of analytic functions. The solution to the well-known Paley problem on the growth of entire functions of finite order is given as one possible application. Series: Operator Theory: Theory and Applications, Volume 67 Contents: INTRODUCTION PREFACE PART I GENERAL PROPERTIES OF ANALYTIC AND FINITE ORDER FUNCTIONS IN THE HALF-PLANE 1 Definition of order and indicator of a function holomorphic in an angle. Relations between various definitions of order 2 Generalized Nevanlinna and Carleman formulas 3 Canonical representation of a function of finite order in the half-plane Necessary Conditions of Completely Regular Growth in the Half-Plane 4 Definition of completely regular growth in the half- plane. List of results on completely regular growth 5 Relation between completely growth in open and closed angles 6 Asymptotic behavior of the modulus and zero distributions of entire functions of the class A (subscript) 7 Existence of argument boundary density for the zero set of a function of the class A (subscript) 8 Existence of boundary and argument densities for the zero set of a function from A (subscript) Sufficient Conditions of Completely Regular Growth in the Half-Plane and Formulas for Indicators 9 The growth of some auxiliary functions of non-integer order 10 A criterion for a function to belong to the class A (subscript), p being non-integer 11 A criterion for a function to belong to the class A (subscript), p being non-integer 12 The argument-boundary symmetry of the zero set of a function of the class A (subscript), p being integer 13 The growth of some auxiliary functions of integer order 14 A criterion for a function to belong to the class A (subscript), p integer 15 A criterion for a function to belong to the class A (subscript), p integer 16 Functions of the class A (subscript) for even and for odd p 17 Functions of a finite degree in the half-plane PART II RIEMANN BOUNDARY PROBLEM WITH AN INFINITE INDEX WHEN THE VERTICITY INDEX IS LESS THAN 1/2 18 Statement of the homogeneous problem 19 Canonical function 20 Solution of the homogenous problem in the class BL(subscript) . Description of solutions of order p 21 Formulation of the non-homogenous problem and an approach to its solution 22 Solution of the non-homogeneous problem Riemann Boundary Problem with Infinite Index in the Case of Verticity of Infinite Order 23 Statement of the homogeneous problem 24 Canonical function 25 Asymptotic properties of zero sets solutions of the homogeneous problem from the classes B and B a(subscript) 26 General form of solutions of the homogeneous problem in the class B 27 General form of solutions of the homogeneous problem in the class B a (subscript) . 28 An example of a solution of the homogeneous problem in the class B(subscript) a . Importance of the restriction on the exponent in the Holder condition for the function (t) = arg G (t)/(2pit p (superscript)) 29 Statement of the non-homogeneous problem and an approach to its solutions 30 Auxiliary statements 31 Solution of the homogeneous problem Riemann Boundary Problem with a Negative Index 32 An example of a solvable homogeneous problem with a negative index 33 Conditions of unsolvability of the homogeneous problem with a negative index 34 Conditions of solvability of the non-homogeneous problem with an index On the Paley Problem A.1 Formulation of the problem and proof of the main inequality A.2 Solution of the Paley problem Bibliography

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