Best approximations of periodic functions in generalized lebesgue spaces

N p < 1; then the spaces L / are a special case of the so-called Musielak–Orlicz spaces [3]. For the first time, Lebesgue spaces with variable exponent were introduced by Orlicz [4]. In [5], the spaces L / were regarded as an example of more general functional spaces. Later, they were investigated by numerous authors in different directions. For the presentation of the main results obtained in the theory of these spaces, see, e.g., [1, 2, 6–9]. We also note that the generalized Lebesgue spaces with variable exponent are used in the theory of elasticity, mechanics, theory of differential operators, and variational calculus [10–12]. We now present some definitions and known facts required for our subsequent presentation. An important role in the theory of the spaces L / is played by the so-called Dini–Lipschitz condition for the exponent p D p.x/: