Trigonometric aproximation and function spaces with generalized smoothness

• Applicant: Professor Dr. Hans-Jürgen Schmeißer
• Subject Area: Mathematics
• Term: from 2013 to 2014
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 241673541

During the last years in a series of joint papers with Dr. K. Runovski a unified approach to approximation of functions in periodic Lebesgue spaces for the full range of parameters p between zero and infinity has been developed. It relies on the introduction of families of polynomial linear operators and their use as constructive approximation method. Classical approximation processes are contained as special cases. The key result is a General Equivalence Theorem. It means a general theorem on the equivalence of approximation error in an appropriate (quasi) metric, polynomial.K-Functional, generated by an associated differential operator, and modulus of smoothness, generated by an associated periodic function. Hereby it turned out that the equivalence of approximation error and modulus requires an adapted modulus of smoothness. According to the asymptotic behaviour of the modulus of smoothness new scales of function spaces of Sobolev-Besov type generalizing the classical spaces are introduced. The main goal of our research is to study these new smoothness spaces and their interrelations with corresponding approximation processes.We want to determine saturation order and saturation classes of approximation processes, polynomial K-functionals and generalized moduli of smoothness. Moreover, the characterization of function spaces via approximation processes, necessary and sufficient conditions for continuous and compact embeddings of Sobolev type will be investigated. Our method is based on the use of the above mentioned general equivalence theorem. New spaces and classical spaces are compared.The long-term successful cooperation with Dr. K. Runovski has been supported several times by German Research Foundation (DFG) as well as by AVH-Foundation. The present project serves also as starting point for cooperation of our research group with specialists from the Sevastopol branch of the Moscow State University in a field connected with the study of oscillations in various mathematical and physical problems.

• DFG Programme: Research Grants