Nonlinear Approximation of Signals with Respect to Orthonormal Bases
Prof. Martin Gevorg Grigoryan
Yerevan State University, Armenia
Abstract: The lecture is dedicated to nonlinear approximation of signals with respect to orthonormal bases (in particular, the trigonometric system and the Walsh system).
For a given orthonormal basis and increasing sequence new classes of signals will be defined and during the lecture questions about the rate of the best m-term approximations with respect to the given basis will be discussed.
Brief Biography of the Speaker: Martin Gevorg Grigoryan from 1971 to 1976, he studied at and graduated from the Faculty of Mathematics and Mechanics of Yerevan State University (YSU). From 1976 to 1979, he pursued postgraduate studies at YSU. In 1979, he defended his PhD thesis on “On the Convergence of Orthogonal Series in Metrics” under the supervision of A.A. Talalyan. Later, in 1997, he defended his doctoral dissertation on the topic “ Convergence of Fourier series in the complete orthogonal systems”.
Grigoryan began teaching Mathematical Analysis at YSU during his fifth year as a student. From 1983 to 2000, he served as an associate professor, and from 2000 to 2002, as a professor. Since 2002, he has been the head of the Department of Higher Mathematics at YSU.
In 2012, Grigoryan established and founded the Laboratory of Linear and Nonlinear Approximations and Their Applications at YSU, which he continues to lead. Under his supervision, more than 15 doctoral and candidate theses have been defended. Among his students are Sergo Episkoposyan, Smbat Gogyan, Artsrun Sargsyan, and Levon Galoyan. His disciples Artavazd Minasyan, Stepan Sargsyan, Arthur Kobelyan, and Levon Galoyan are the core research team of the internationally renowned tech company KRISP, with Artavazd Minasyan also being a co-founder of KRISP.
Grigoryan has authored over 180 articles on the following topics:
Convergence of ordinary and double Fourier series in classical and general orthonormal systems,
Linear and nonlinear approximations, Greedy and summation methods, Convergence of Cesàro means, Universal series, Universal functions.
Grigoryan defined universal functions (in different senses) for various classes of functions with respect to the classical systems and in this direction obtained a number of important results, in particular:
He constructed an integrable function whose Fourier-Walsh series converges almost everywhere and which is an almost universal function for the class of finite measurable functions with respect to the Walsh system.
Grigoryan proved that by modifying the values of any integrable signal on a set of small measure, it can become an almost universal signal for the class of measurable signals with respect to the trigonometric system.
In Yerevan, Armenia, July 1 - 6, 2024. was organised jointly by Ghent Analysis and PDE Center and Yerevan State University, and is dedicated to the 70th anniversary of Prof. Martin Grigoryan
(see https://www.gmg70.com).