Contemporary Approaches to Approximation of Operators

This carefully curated volume presents a comprehensive study of approximation processes for functions and operators, bringing together classical techniques and modern developments in the field. The focus is on linear and nonlinear approximation methods, with particular emphasis on operator theory and its applications. Serving as both a reference and a research monograph, the collection is intended for graduate students, researchers, and practitioners working in approximation theory, operator theory, and several applied sciences. Special attention is paid to the approximation of functionals, such as integrals, i.e., to various aspects of quadrature and cubature formulas, including their applications in numerical integration, solving integral equations, and so on. Moreover, the book explores polynomial, trigonometric, and spline approximations, construction of operators, composition of operators before advancing to operator-based methods such as Bernstein, Kantorovich, Durrmeyer, and their generalizations. Emphasis is given to moment estimates, asymptotic expansions, probabilistic approaches, sampling theory and connections with special functions, neural networks and orthogonal polynomials. The discussion develops tools including moment estimates, asymptotic expansions, and probabilistic methods, while also addressing approximation in weighted spaces, exponential-type operators, and connections with orthogonal polynomials. Attention is given to complete asymptotic expansions and the interplay between constructive approximation and operator theory.