In the present work we prove some direct and inverse theorems for approximation by trigonometric polynomials in Musielak–Orlicz spaces. Furthermore, we get a constructive characterization of the Lipschitz classes in these spaces. | Turk J Math (2018) 42: 2166 – 2185 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Convolution and Jackson inequalities in Musielak–Orlicz spaces 1 Ramazan AKGÜN1 ,,Yunus Emre YILDIRIR2,∗, Department of Mathematics, Faculty of Arts and Science, Balıkesir University, Balıkesir, Turkey 2 Department of Mathematics, Faculty of Education, Balıkesir University, Balıkesir, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: In the present work we prove some direct and inverse theorems for approximation by trigonometric polynomials in Musielak–Orlicz spaces. Furthermore, we get a constructive characterization of the Lipschitz classes in these spaces. Key words: Musielak–Orlicz space, direct and inverse theorem, Lipschitz class, trigonometric approximation 1. Introduction Musielak–Orlicz spaces are similar to Orlicz spaces but are defined by a more general function with two variables φ (x, t) . In these spaces, the norm is given by virtue of the integral ∫ φ (x, |f (x)|) dx, T where T := [−π, π]. We know that in an Orlicz space, φ would be independent of x, φ (|f (x)|) . The special cases φ (t) = tp and φ (x, t) = tp(x) give the Lebesgue spaces Lp and the variable exponent Lebesgue spaces Lp(x) , respectively. In addition to being a natural generalization that covers results from both variable exponent and Orlicz spaces, the study of Musielak–Orlicz spaces can be motivated by applications to differential equations [13, 28], fluid dynamics [15, 23], and image processing [5, 10, 16]. Detailed information on Musielak–Orlicz spaces can be found in the book by Musielak [26]. Polynomial approximation problems in Musielak–Orlicz spaces have a long history. Orlicz spaces, which satisfy the translation invariance property, are a particular case of Musielak–Orlicz spaces. In these spaces, polynomial approximation problems were investigated by .
