THE THEORY OF LINEAR OPERATORS FROM THE STANDPOINT OF DIFFEREN TIAL EQUATIONS OF INFINITE ORDER By HAROLD T. DAVIS INDIANA UNIVERSITY AND THE COWLES COMMISSION FOR RESEARCH IN ECONOMICS THE PRINCIPIA PRESS Bloommgton, Indiana 1936 MONOGRAPH OF THE WATERMAN INSTITUTE OF INDIANA UNIVERSITY CONTRIBUTION NO. 72 THE THEORY OF LINEAR OPERATORS To Agnes, who endured so patiently the writing of it, this boo is affectionately dedicated. TABLE OF CONTENTS CHAPTER I LINEAR OPERATORS 1. The Nature of Operators ------------1 2. Definition of an Operator -----.--3 3. A Classification of Operational Methods --------7 4. The Formal Theory of Operators ----------g 5. Generalized Integration and Differentiation - - 16 6. Differential and Integral Equations of Infinite Order -----23 7. The Generatrix Calculus - - 28 8. The Heaviside Operational Calculus ---------34 9. The Theory of Functionals ------------33 10. The Calculus of Forms in Infinitely Many Variables -----4 CHAPTER II PARTICULAR OPERATORS 1. Introduction ----------------51 2. Polynomial Operators --------53 3. The Fourier Definition of an Operator ---------53 4. The Operational Symbol of von Neumann and Stone -----57 5. The Operator as a Laplace Transform ---------59 6. Polar Operators ...-60 7. Branch Point Operators ------------64 8. Note on the Complementary Function ---------70 9. Riemanns Theory - .--.--72 10. Functions Permutable with Unity ----------76 11. Logarithmic Operators ------------78 12. Special Operators --------------85 13. The General Analytic Operator ----------99 14. The Differential Operator of Infinite Order -------100 15. Differential Operators as a Cauchy Integral -------103 16. The Generatrix of Differential Operators--------104 17. Five Operators of Analysis ------------105 CHAPTER III THE THEORY OF LINEAR SYSTEMS OF EQUATIONS 1. Preliminary Remarks -------------108 2. Types of Matrices --------------109 3. The Convergence of an Infinite Determinant -------114 4. The Upper Bound of a Determinant. Hadamards Theorem - - 116 5. Determinants which do not Vanish - - - - - - - - - 123 6. The Method of the Liouville-Neumann Series -------126 7. The Method of Segments ------------130 8. Applications of the Method of Segments. --------132 9. The Hilbert Theory of Linear Equations in an Infinite Number of Variables - - - - 137 10. Extension of the Foregoing Theory to Holder Space 149 vii Vlll THE THEORY OF LINEAR OPERATORS CHAPTER IV OPERATIONAL MULTIPLICATION AND INVERSION 1. Algebra and Operators -------.. --153 2. The Generalized Formula of Leibnitz ---------154 3. Bourlets Operational Product --. 155 4. The Algebra of Functions of Composition --------159 5. Selected Problems in the Algebra of Permutable Functions - - - - 164 G. The Calculation of a Function Permutable with a Given Function - 166 7. The Transformation of Peres -----------171 8. The Permutability of Functions Permutable with a Given Function - 173 9. Permutable Functions of Second Kind - --176 10. The Inversion of Operators Bourlets Theory ------177 It. The Method of Successive Substitutions --------181 12. Some Further Properties of the Resolvent Generatrix - 185 13. The Inversion of Operators by Infinite Differentiation - 188 14. The Permutability of Linear PilYeiential Operators -----190 15. A Class of Non-permutable Operators ---------194 16. Special Examples Illustrating the Application of Operational Processes 200 CHAPTER V GRADESDEFINED BY SPECIAL OPERATORS 1. Definition ----------------211 2. The Grade of an Unlimitedly Differentiable Function - 212 3. Functions of Finite Grade ------------215 4. Asymptotic Expansions --- 222 5. The Summability of Differential Operators with Constant Coefficients 230 6. The Summability of Operators of Laplace Type ------235 CHAPTER VI DIFFERENTIAL EQUATIONS OF INFINITE ORDER WITH CONSTANT COEFFICIENTS 1. Introduction ---------------238 2. Expansion of the Resolvent Generatrix --------239 3. The Method of Cauchy-Bromwich ----------250 4...