In this paper we present the definition and some properties of (IC)-a.p. functions, i.e. uniformly almost periodic (IC)(n)-a.p. functions with their indefinite integrals. Next, we give the definition and some properties of (IC)(n) -a.p. functions, i.e. uniformly almost periodic functions with their n derivatives and indefinite indegrals, and (IC)(n) -a.p. functions, i.e. uniformly almost periodic functions with their every derivatives, with respect to a positive sequence a = (a1), and indefinite integrals.
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In this note we present remarks on completeness of spaces of almost periodic functions of various types and some theorems on connection between almost periodicity of the derivative and the indefinite integral of the function and almost periodicity of the function.
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The paper gives a theorem on the indefinite integral of H-almost periodic function and a theorem on approximation of an H-almost periodic function by means Steklov functions.
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