This is a survey of the various definitions and characterizations of Bohr-, Stepanoff-, Weyl- and Besicovitch almost periodic functions and their relations, some related classes and generalizations, with some (counter)examples. Such an attempt is laudable, and the list of 137 references is certainly valuable for anyone interested in this field. The representation however is suboptimal: The notation of the authors is different from the standard one introduced by Besicovitch, Rohr et al., it is not consistent and sometimes misleading. The definitions/theorems 2.8/2.11, 3.4/3.5, 4.6/4.11, 5.5 are formally not correct, since all the spaces considered by the authors contain only real-valued functions, whereas their trigonometric polynomials (Remark 2.7) are complex-valued.
The \(e\)-\(W^p\) of Definition 4.2 coincides with the \(W^p\) of Def. 4.6, so the statements “\(e\)-\(W^p= S^p\)” and “\(W^p\nRightarrow e\)-\(W^p\)” of Table 2 (p. 176) and their use in Remark 4.26 and on p. 150 are not correct; also not correct in Table 2 are “\(W^p\)-normal \(\Rightarrow e\)-\(W^p_{ap}\)”(by Example 4.29) and “\(e\)-\(W^p\)-normal \(\Rightarrow e\)-\(W^p\)”. Theorem 7.5 on the countability of the Hartman spectrum seems open, since the result of Kahane cited there works only for the half-line. The essential “\(ap\)” part of Def. 5.13 is missing, in Theorem 6.12 the \(G^2\) should be \(B^2\), in theorem 2.20 it should be \(m_k\in\mathbb{Z}\). Many undefined terms, missing references, vaguely formulated definitions and various misprints make the reading unnecessarily difficult.