## "Babes-Bolyai" University of Cluj-Napoca Faculty of Mathematics and Computer Science

 Special topics of function theory
 Code Semes-ter Hours: C+S+L Credits Type Section MT005 6 2+1+0 5 optional Matematică MT005 6 2+1+0 5 optional Matematică
 Teaching Staff in Charge
 Assoc.Prof. SERB Ioan Valeriu, Ph.D., ivserb@math.ubbcluj.roProf. NEMETH Alexandru, Ph.D., nemab@math.ubbcluj.ro
 Aims The purpose of this course is the appropriation of the main techniques and knowledges in the theory of duality for clases of convex functions and applications. The main objectifs are: the study of complementary pairs of N-functions, the construction of Orlicz spaces and the study of the geometry of Orlicz spaces.
 Content Chapter I. Special classes of convex functions 1. N-functions. Convex functions. Integral reprezentation of convex functions. First definition of N-functions. Properties of N-functions. The second definition of N-functions. 2. The complementary function of a N-function. Young inequality for N-functions. Transmission of inequalities to complementary functions of N-functions. 3.Comparison between classes of N-functions. Comparisons between the complementary functions of N-functions. The main part of N-functions. A sufficient condition for equivalence of two functions. The construction of new classes of equivalence. 4. Delta-2 condition. N-functions verifyinf the Delta-2 condition. Expression of Delta-2 condition in terms of complementary functions of N-functions. Chapter II. Orlicz spaces 1. Orlicz classes of measurable functions. The definition of Orlicz classes. Comparisons between Orlicz classes and L_1 or L_infinit. Integral inequality of Jensen. Comparisons between Orlicz classes. Algebraic structure of Orlicz classes. 2. Orlicz spaces L_M. Orlicz norm. Complete Orlicz spaces. Holder inequality for pairs of Orlicz spaces. Gauge norm on L_M. Comparisons between Orlicz spaces. Orlicz and gauge norm of characteristic function. 3. The adjoint space of L_M. Linear and continuous functionals, their general form and their norm. Reflexivity of Orlicz spaces.
 References 1. Krasnoselskii, M.A., Rutickii, Ya.B., Functii convexe si spatii Orlicz (l. rusa) 1959. 2. Rao, M.M.and Ren Z.D., The Theory of Orlicz spaces (l. engleza) Marcel Decker Inc., New York, 1991. 3. Birnbaum, Z.W., Orlicz, W., Uber die Veralgemainerung des Begriffes der zueinander konjugierten Potenzen, Studia Math. 3 (1931) 1-67. 4. Salehov, D.V., Despre norma functionalelor liniare si continue in spatii Orlicz si despre o caracteristica a spatiilor L_p. (l. rusa) D.A.N. SSSR 111, 5 (1956). 5.Kantorovici, L.V., Akilov, G.P., Funcional analysis, Pergamon Press, Inc., New York, 1982. 6. Maleev, R.P., Troyanski, S.L., On the moduli of convexity and smoothness in Orlicz spaces, Studia Math. 54, (1975), 131-141. 7. Figiel, T., On the moduli of convexity and smoothness, Studia Math. 56 (1976), 121-155.
 Assessment Exam.