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| Ever since their introduction by H.~Bohr in mid-twenties, almost periodic functions of various types have been extensively studied by mathematicians. During the talk we are going to pay close attention to a certain class of not necessarily bounded almost periodic functions, which contains the class $AP(\mathbb X)$, namely to the class of almost periodic functions in the sense of Levitan. Apart from discussing their basic properties, we are going to provide several results concerning convolution of Levitan almost periodic functions as well as necessary and sufficient conditions for a nonautonomous superposition operator to map the class of such functions into itself. Let us mention that the latter problem is closely connected with the question about natural topology on the class $LAP$. Furthermore, we are going also to discuss a new result concerning the asymptotic behaviour of certain Levitan almost periodic functions, whose proof required a completely new approach via continued fractions. Finally, we are going to indicate applications of the presented results to differential and integral equations. |
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