Nonlinear approximations of classes of functions with bounded mixed derivative in Lorentz space

Project manager: Myrzagalieva A. H.

Funding source: GF of young scientists for the project “Zhas Galym”

Years of implementation: 2024–2026

Amount of funding: 29,929,450 tenge

• Study of nonlinear approximation properties of a class of differentiable functions of many variables in Lorentz space;
• Study of approximation properties of trigonometric and linear diameters of a class of functions with mixed derivative in Lorentz space;
• Finding the exact order of the best M-term approximation by trigonometric polynomials of the generalized Sobolev space L_q^ψ in Lorentz space.

• Obtaining the exact order of the best M-term approximation by trigonometric polynomials of the Sobolev space W_(q,τ_1)^r ̅ in the Lorentz space L_(p,τ_2 ) (T^m ) for various ratios of parameters q,p,τ_1,τ_2∈[1,∞] and r_j>0,j=1,…,m;
• Finding the exact order of the best M-term approximation of the multiple Bernoulli kernel in the Lorentz and Marcinkiewicz space;
• Obtaining the exact order of the trigonometric and linear diameters λ_n (W_(q,τ_1)^r ̅ )_(p,τ_2 ) of the Sobolev class W_(q,τ_1)^r ̅ in the Lorentz space for various ratios of parameters q,p,τ_1,τ_2∈[1,∞] and r_j>0,j=1,…,m;
• Obtaining the exact order of the best M-term approximation by trigonometric polynomials of the generalized Sobolev space L_q^ψ in the Lorentz space.

• The Project will find new nonlinear approximative properties of functional classes of differentiable functions of many variables, generalized classes of functions, as well as trigonometric, linear diameters of functional compacts, determine their exact order of the best M-term approximation for various parameter ratios, as well as the multiple Bernoulli kernel in the Lorentz and Marcinkiewicz space.
• All results of the Project will be provided with rigorous mathematical proofs and tested at scientific seminars and international conferences.

• New nonlinear approximation properties of functional classes of differentiable functions of many variables were found, their exact order of the best M-term approximation was determined for various parameter ratios, namely, upper bounds for the best M-term approximation by trigonometric polynomials of the Sobolev space in the Lorentz space were obtained for various parameter ratios.
• Lower bounds for the best M-term approximation by trigonometric polynomials of the Sobolev space in the Lorentz space were obtained for various parameter ratios.
• Methods based on a combination of results from the theory of approximations with harmonics from hyperbolic crosses and modern results on greedy algorithms were used and developed. The main method of proof in a number of theorems was the idea of ​​representing a function in the form of blocks of harmonics, based on a theorem of the type of the Littlewood-Paley theorem, the use of inequalities such as the Bernstein inequality, Nikol’skii inequality, Marcinkiewicz’s theorem on multipliers, as well as inequalities connecting the norms of functions in different metrics, such as the Temlyakov inequality and others..​