Approximation theory

Faculty members involved

Kirill Kopotun (link) has particular interest in various topics in nonlinear approximation, constrained approximation, measures of smoothness, weighted approximation, adaptive algorithms, and other related areas.

One of the central tasks in Approximation Theory is to determine a connection between approximation and smoothness properties of what's been approximated (functions, surfaces, etc.), and the main goal of the research program is further investigation of the exact connection between smoothness of (classes of) functions (surfaces, bodies, etc.) and their approximation orders. In general, nonlinear approximation methods perform much better than linear ones, and the order of approximation from nonlinear manifolds is much better than the order of approximation by the elements of linear spaces, which is the main reason for investigating nonlinear approximation techniques.

Andriy Prymak (link) focuses on rather theoretical aspects of approximation theory, particularly on shape-preserving approximation, and on relations of measures of smoothness and approximation. Problems in shape-preserving approximation demand the approximating tool (often a polynomial or a spline) to possess the same geometric property (monotonicity, convexity) as the original function does. In many cases one can achieve the same order of approximation as in the unconstrained case, but this may require very different methods, especially in the multivariable settings. Rate of approximation can be related to certain measures of smoothness of a function (more sensitive analogs of derivatives). His work in this direction includes generalizations of various classical inequalities, connections of approximation theory to geometry of Banach spaces, and studies of "boundary effects" in approximation on convex domains. His other interests include various properties of convex functions and bodies, relations of approximation theory to other areas of analysis (harmonic analysis and Fourier analysis) and mathematics (graph theory), computational and numerical methods.

Current students and postdoctoral fellows

  • Isiaka Akinsanya (M.Sc.) - K. Kopotun
  • Farzaneh Jannat (M.Sc.) - K. Kopotun
  • Kyrylo Muliarchyk (M.Sc.) - K. Kopotun
  • Sahar Rahimzad Lamey (M.Sc.) - K. Kopotun
  • Olena Usoltseva (Ph.D.) - A. Prymak

Representative recent publications

  • A. Bondarenko, A. Prymak, D. Radchenko, On concentrators and related approximation constants, J. Math. Anal. Appl., 402 (2013) 234-241
  • K. A. Kopotun, D. Leviatan, A. Prymak, I. A. Shevchuk, Uniform and pointwise shape preserving approximation by algebraic polynomials, Surv. Approx. Theory, 6 (2011), 24-74
  • Z. Ditzian, A. Prymak, Convexity, moduli of smoothness and a Jackson-type inequality, Acta Math. Hung., 130 (2011), no. 3, 254-285
  • Z. Ditzian, A. Prymak, Approximation by dilated averages and K-functionals, Canadian Journal of Mathematics, 62 (2010), no. 4, 737-757

News

MathCamp 2017 information is online.

Manitoba Workshop on Mathematical Imaging Science Friday, May 5, all day, Robert Schultz Lecture Theatre, details.

In an effort to help students, the Math department has put together the LevelUp program. See details here. Video explaining registration process is here.

Events

Thursday, May 4th, 2017 at 15:30, 418 Machray Hall
Tommy Kucera
Fibonacci and The Liber Abaci
(Seminar series : Colloquium)

Friday, May 12th, 2017 at 15:30, 418 Machray Hall
John Dallon
Modeling Amoeboidal Cell Motion -- Force vs Speed
(Seminar series : Colloquium)

Friday, August 4th, 2017 at 15:30, 418 Machray Hall
Jose Aguayo
TBA
(Seminar series : Colloquium)