This book presents solutions to various extremal problems in analysis and provides mathematical tools that can be used to solve complex problems in different areas of fundamental and applied science. These problems are not just theoretical, but they have practical applications in various fields including optimization and image processing. The book includes an in-depth exploration of Ostrowski type inequalities, covering their applications to operator theory, inequalities for derivatives, and optimization of cubature formulae. The authors also provide a comprehensive treatment of the Bojanov-Naidenov and Erdos problems and investigate their solutions for various function classes including differentiable functions, trigonometric polynomials, and splines. Additionally, the book explores sharp Remez type inequalities. These inequalities are examined in different metrics for various function classes including trigonometric polynomials and splines. The authors also include innovative research in image processing, focusing on the restoration of noise-corrupted optical images with simultaneous contrast enhancement and a variational approach to simultaneous fusion and denoising of color images with different spatial resolutions. This book is a valuable resource for researchers and graduate students in approximation theory, numerical analysis, and image processing.

Autorentext

Vladyslav Babenko, Ph. D., Habilitated Doctor in Mathematics, is Professor in the Department of Mathematical Analysis and Optimization at Oles Honchar Dnipro National University, where his teaching focuses on functional analysis and theory of Functions of real and complex variables. His research interests include areas of analysis and approximation theory. Dr. Babenko has over 450 journal publications in mathematical sciences and has published four books.

Vladimir Kofanov, Ph. D., Habilitated Doctor in Mathematics, is Professor in the Department of Mathematical Analysis and Optimization at Oles Honchar Dnipro National University, where his teaching focuses on functional analysis and theory of functions of real and complex variables. Hi research interests include different areas of theory of inequalities for derivatives of differentiable functions as well as polynomials and splines and their applications in approximation theory. Dr. Kofanov has over 200 journal publications in mathematical sciences and has published three books.

Peter Kogut, Ph. D., Habilitated Doctor in Mathematics, is Professor in the Department of Mathematical Analysis and Optimization at Oles Honchar Dnipro National University, where his teaching focuses on variational methods in mathematical physics, optimal control theory for distributed systems, and nonlinear functional analysis. His research interests include the different areas of optimization theory, optimal control problems for PDEs, image processing, asymptotic analysis, optimization in partially ordered spaces, and homogenization theory. He has over 200 journal publications in mathematical sciences and two books.

Oleg Kovalenko, Ph. D., Habilitated Doctor in Mathematics, is Professor in the Department of Mathematical Analysis and Optimization at Oles Honchar Dnipro National University, where his teaching focuses on mathematical analysis, complex analysis, and numeric Methods. He conducts research in the following areas: extremal problems in approximation theory for numeric and non-numeric functions, inequalities for derivatives, and optimal recovery problems.

Nataliia Parfinovych, Ph. D., Habilitated Doctor in Mathematics, is Head of the Department of Mathematical Analysis and Optimization at Oles Honchar Dnipro National University, where her teaching focuses on mathematical analysis, approximation theory, harmonic analysis and its application, and mathematics teaching methods. She conducts research and has achieved notable results in the following areas: extremal problems in approximation theory, approximation of individual functions and functional classes by splines and their analogues, investigation of internal properties of approximating aggregates, inequalities for norms of fractional and integer derivatives, approximation of unbounded operators by bounded ones, and optimal recovery of operators on inaccurate information. She is the Editor-in-Chief of the scientific journal Research in Mathematics.