Institut für Mathematik » Mitarbeiter

Approximation of Functions and Fourier Multipliers and their applications - AFFMA

1 November 2016 — 31 October 2018

This project has received funding from the European Union's Horizon 2020 research and innovation programm under the Marie Skłodowska-Curie grant agreement No 704030.

Beneficiary:

Universität zu Lübeck, Institut of Mathematics
Supervisor: Prof. Dr. Jürgen Prestin

Partner Organisations:

Centre de Recerca Matemàtica: CRM
Lead Scientist: Prof. Dr. Sergey Tikhonov
Bar-Ilan University, Department of Mathematics
Lead Scientist: Prof. Dr. Elijah Liflyand

Objective:

This research project is concerned with the following three topics in approximation theory and Fourier analysis:

Simultaneous approximation of functions and their derivatives in L_p, 0<p<1. We expect to investigate classes of functions and different methods of approximation for which the problem of simultaneous approximation is solvable, and to obtain estimates for the errors of the best approximation of functions and their derivatives for particular methods of approximation in L_p, 0<p<1.
New inequalities for moduli of smoothness of functions in L_p, 0<p<1. We expect to find the classes of functions for which the direct and inverse inequalities for moduli of smoothness of functions and their derivatives hold, and to investigate the sharp Ulyanov inequality for different concepts of smoothness.
Fourier multipliers and families of multiplier operators in L_p, p>0. We expect to obtain sufficient conditions of the boundedness for such operators in terms of the simultaneous behavior of a multiplier and its derivatives in different functional spaces, and to apply such conditions for solving problems from this proposal.

Publications:

Yu. Kolomoitsev, T. Lomako, J. Prestin On L_p-error of bivariate polynomial interpolation on the square J. Approx. Theory 229 (2018), 13-35
Yu. Kolomoitsev On moduli of smoothness and averaged differences of fractional order Fract. Calc. Appl. Anal. 20 (2017), no. 4, 988-1009
Yu. Kolomoitsev, E. Liflyand On weighted conditions for the absolute convergence of Fourier integrals J. Math. Anal. Appl. 456 (2017), no. 1, 163-176
Yu. Kolomoitsev Best approximations and moduli of smoothness of functions and their derivatives in L_p, 0<p<1 J. Approx. Theory 232 (2018) 12-42
Yu. Kolomoitsev, M. Skopina Approximation by multivariate Kantorovich-Kotelnikov operators J. Math. Anal. Appl. 456 (2017), no. 1, 195-213
Yu. Kolomoitsev, T. Lomako Inequalities in Approximation Theory Involving Fractional Smoothness in L_p, 0<p<1 In: Abell M., Iacob E., Stokolos A., Taylor S., Tikhonov S., Zhu J. (eds) Topics in Classical and Modern Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham, 2019, pp 183-209
Yu. Kolomoitsev, S. Tikhonov Hardy-Littlewood and Ulyanov inequalities to appear in Mem. Amer. Math. Soc.
Yu. Kolomoitsev, M. Skopina Around Kotelnikov-Shannon formula Conference: 2017 International Conference on Sampling Theory and Applications (SampTA), Tallin, Estonia, 2017, pp. 279-282

Seminars:

Talk "Best approximations and moduli of smoothness of functions and their derivatives in L_p, 0<p<1", Institute of Mathematics NAS of Ukraine, Kiev, Ukraine, February 3, 2017.
Talk "Best approximations and moduli of smoothness of functions and their derivatives in L_p, 0<p<1", Holon Institute of Technology, Holon, Israel , May 14, 2017.
Talk "On the growth of Lebesgue constants for convex polyhedra", Bar-Ilan University, Ramat-Gan, Israel, May 15, 2017.
Talk "Best approximations and moduli of smoothness of functions and their derivatives in L_p, 0<p<1", Technion - Israel Institute of Technology, Haifa, Israel, May 21, 2017.
Talk "Approximation by multivariate Kantorovich-Kotelnikov sampling operators", Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, January 12, 2018.
Talk "Approximation by generalized Kantorovich-Kotelnikov sampling operators", Seminari d'Anàlisi de Barcelona, March 05, 2018.
Talk "On the growth of Lebesgue constants for convex polyhedra", Institute of Mathematics, University of Giessen, Germany, April 26, 2018.
Talk "On the growth of Lebesgue constants for convex polyhedra", Universität Osnabrück, Germany, 2 May 2018.
Talk "On approximation by multivariate Kantorovich-Kotelnikov sampling operators", Technische Universität Chemnitz, Germany, 8 May 2018.
Talk "On approximation by multivariate Kantorovich-Kotelnikov sampling operators", Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austria, June 1, 2018
Talk "Sharp Ulyanov inequalities in Lebesgue spaces", Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Ukraine, July 16, 2018

Conferences:

Mecklenburg Workshop Approximation Methods and Fast Algorithms, Hasenwinkel, Germany, September 10- September 14, 2018.
9th International Conference on Curves and Surfaces, Arcachon, France, June 28 to July 4, 2018.
Harmonic Analysis and Applications (Strobl 18), Strobl, Austria, June 4 - June 8, 2018.
New perspectives in the theory of function spaces and their applications (NPFSA-2017), Będlewo, Poland, September 17 - September 23, 2017.
6th Workshop on Fourier Analysis and Related Fields, Pécs, Hungary, August 24 - August 31, 2017.
11th International ISAAC Congress, Växjö, Sweden, August 14 - August 18, 2017
Summer School in Lübeck "Modeling, Analysis, and Approximation Theory toward applications in tomography and inverse problems", Lübeck, Germany, July 31 - August 4, 2017.
Foundations of Computational Mathematics (FoCM 2017) Barcelona, Spain, July 10 - July 19, 2017.
Follow-up: Approximation Theory & Function Spaces, Barcelona, Spain, June 26 - June 30, 2017.
Complex and Harmonic Analysis III, Holon and Haifa, Israel, June 04 - June 08, 2017.
3rd International Conference "Approximation Methods for Molecular Modelling and Diagnosis Tools", Kyiv, Ukraine, January 26 - January 30, 2017.

Scientific visits:

Centre de Recerca Matemàtica, Barcelona, Spain, February 9 - March 2, 2017.
Bar-Ilan University, Ramat-Gan, Israel, May 2 - June 18, 2017.
Centre de Recerca Matemàtica, Barcelona, Spain, June 25 - July 13, 2017.
Laboratoire d'Analyse et de Mathématiques Appliquées, University Paris-Est, Paris, France, October 14 - October 29, 2017.
Centre de Recerca Matemàtica, Barcelona, Spain, February 28 - March 20, 2018.
Centre de Recerca Matemàtica, Barcelona, Spain, October 4 - October 25, 2018..