57 BOGOLYUBOV 



the Bologna Academy of Sciences. A number of the investi- 

 gations by Bogolyubov have dealt with the theory of quasi - 

 periodic functions. He showed that the basic theorems of quasi- 

 periodic functions (for instance, the theorem of the uniform 

 approximation of a continuous quasi -periodic function by trigo- 

 nometric sums) result from one general theorem in the field of 

 an arbitrary limited function. According to this theorem, 

 certain linear combinations from an arbitrary limited function 

 are capable of being approximated by trigonometric sums. The 

 proof of the approximation theorem for the quasi -periodic 

 functions of Bohr, presented by Bogolyubov, does not rely upon 

 the Parseval equality; in general, it relies upon virtually none 

 of the properties of functions quasi-periodic in the sense of 

 Bohr. In the proof of this theorem, the underlying principle is 

 an original purely mathematical conception of the properties of 

 quasi -periods. In this Bogolyubov has presented a virtually new 

 synthesis of Bohr's theory of quasi -periodic functions. 



Bogolyubov has carried out a series of investigations dealing 

 with the theory of differential equations with limiting con- 

 ditions, directly linked to the application of the differentiation 

 method to the calculus of variations. The basis of these in- 

 vestigations is the estimation of error in the approximate de- 

 termination of proper values and characteristic functions of the 

 boundary. The approximation method developed here by 

 Bogolyubov is applicable not only to the solution of boundary 

 problems, but also to the solution of partial differential 

 equations. Starting in 1932, he began work with N. M. Krilov 

 on the development of a completely new branch of mathematical 

 physics—the theory of nonlinear oscillations which they called 

 nonlinear mechanics. It should be noted that, in the twenties, 

 the rapid development of radio and electrical engineering re- 

 quired a study of nonlinear oscillations. The use, for this pur- 

 pose, of methods developed by A. Poincare and A. M. Lyapunov 

 was completely inadequate. It was necessary to develop new, 

 more flexible methods of investigation of all the complex phe- 

 nomena originating in nonlinear oscillatory systems. The re- 

 search of Bogolyubov developed in two principal directions: that 

 of the development of methods for the asymptotic integration of 

 nonlinear equations describing oscillatory processes, and that 

 of the mathematical substantiation of these methods, and this 

 was equivalent to the development of a general theory of dy- 

 namic systems. 



In the first of these directions, having to do with the study 

 of differential equations with a "small" or "large" parameter. 



