Investigation of the Approximation of Continuous Periodic Functions on the Torus

Abstract


 
 
 Main purpose of the present work is development of qualitative theory of difference equations in the space of bounded numeric sequences.Main result is the establishment of necessary conditions of the existence of invariant toroidal manifolds for countable systems of differential and difference equations. In order to solve this problem, observed spaces are constructed in a special way. Necessary conditions of the existence of invariant tori for countable systems of differential and difference equations are derived.A concept of a continuous periodic in each variable function with period 2Pi , values of which lie in l2 , is introduced. Spaces, in which observations are made, are constructed in a special way. A theorem on approximation of a function from the corresponding space bytrigonometric polynomials is proven.