The problem of existence and construction of the atomic wavelet system, which consists of infinitely differentiable functions with a compact support, is considered. Formulas for evaluation atomic wavelets are obtained. Examples of applications of atomic wavelets to approximation of some functions are presented. Compactly supported solutions of some functional differential equations and their properties are considered. A new class of atomic functions is introduced. Approximation properties of the linear spaces of finite linear combinations of translates of the atomic functions are presented. Key words: wavelet, functional differential equation, atomic function, atomic wavelet, function with compact support, infinitely differentiable function, nonstationary wavelet system.
