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TBA
Mohammad Ayman Mursaleen (University of Ostrava) Approximation by positive linear operators Abstract: Approximation theory plays a fundamental role in analysis, with the Weierstrass approximation theorem standing as a cornerstone result. This theorem establishes that every continuous function on \([0,1]\) (or every continuous \(2\pi\)-periodic function on \([0,2\pi])\) can be approximated by algebraic (or trigonometric) polynomials. Bernstein's introduction of Bernstein polynomials provided an elegant and constructive proof of this theorem, sparking interest in the development of operators for broader function spaces. The Szász-Mirakjan operators, independently introduced by Szász and Mirakjan, extend this framework to handle functions in \(C[0,\infty)\). Subsequently, various generalizations of Bernstein operators, including Bernstein-Kantorovich and Bernstein-Durrmeyer operators, have been developed to approximate functions in \(L^p[0,1]\) (\(1 \le p < \infty\)). This seminar will explore the theoretical foundations and practical aspects of these operators. The computation of moments and central moments, determining the order of approximation, discussion of basic results and approximation properties, and also a study of estimates and rate of convergence for these operators by employing various useful tools like the Korovkin type theorem, Peetre's K-functional, and asymptotic error constant. The discussion will culminate with a brief introduction to approximation by neural network operators, highlighting their connection to classical approximation methods and their potential applications in modern computational settings. |