ABSTRACT
In this chapter, we recall some facts of calculus and the theory of ordinary differential equations. In particular, we describe properties of averaging operators, Fourier operators, and integral Laplace operators. We also discuss the notion of a weak derivative introduced by S.L.Sobolev in the 1930’s and some fundamental results concerning Sobolev spaces. We use the average method developed in works of V.A.Steklov and S.L.Sobolev. We also present an integral representation of summable functions constructed by S.V.Uspenskii [1, 2]. This representation is based on the use of special averaging functions.
