Extremal Problems and Inequalities of Markov-Bernstein Type for Algebraic Polynomials

Inequalities for polynomials and their derivatives are essential in approximation theory, and this book covers Extremal Problems and Inequalities of Markov-Bernstein Type for Algebraic Polynomials, trigonometric polynomials, and related functions in different metrics, making it useful for graduate and PhD students studying inverse problems.

Format: Paperback / softback
Length: 442 pages
Publication date: 15 February 2022
Publisher: Elsevier Science Publishing Co Inc

Inequalities for polynomials and their derivatives hold immense significance in numerous fields of mathematics, as well as in computational and applied sciences. These inequalities play a fundamental role in approximation theory, serving as powerful tools for understanding and solving complex problems. In this comprehensive text, we delve into the study of inequalities for algebraic polynomials, trigonometric polynomials, and related functions, employing various metrics across different classes of polynomials. Our focus extends to both general polynomial classes and important restrictive classes, providing a comprehensive treatment of these topics. Primarily designed for graduate and PhD students, this book serves as a valuable resource for researchers exploring problems that necessitate derivative estimates. It is particularly valuable for those engaged in the study of inverse problems in approximation theory.

The importance of inequalities for polynomials and their derivatives cannot be overstated. They serve as fundamental tools for analyzing and solving a wide range of mathematical problems, including those in approximation theory, differential equations, and optimization. By studying these inequalities, researchers can gain insights into the behavior of functions, establish necessary conditions for optimality, and develop efficient algorithms for solving complex problems.

One of the key applications of inequalities for polynomials and their derivatives is in approximation theory. Approximation theory is a branch of mathematics that studies the error bounds for approximating functions by polynomials or other rational functions. Inequalities play a crucial role in this field by providing upper and lower bounds on the error incurred in the approximation. For example, the Cauchy-Schwarz inequality states that the norm of the difference between two polynomials is bounded by the sum of their norms, which allows for the development of efficient algorithms for polynomial approximation.

In addition to approximation theory, inequalities for polynomials and their derivatives are also used in the study of differential equations. Differential equations are equations that involve the derivatives of a function with respect to its variables. By studying inequalities for differential equations, researchers can obtain information about the behavior of solutions, including stability, uniqueness, and existence. For instance, the Hopf-Lax theorem provides a condition for the stability of a solution to a differential equation, which is essential for the analysis of many physical systems.

Furthermore, inequalities for polynomials and their derivatives are important in optimization problems. Optimization is the process of finding the best solution to a problem given a set of constraints. Inequalities play a role in this process by providing conditions that restrict the possible solutions and guide the search for the optimal solution. For example, the Lagrange multiplier method is a technique for solving optimization problems that involves finding the optimal values of Lagrange multipliers to satisfy a set of inequality constraints.

In conclusion, inequalities for polynomials and their derivatives are essential tools for mathematicians, scientists, and engineers. They play a fundamental role in approximation theory, differential equations, and optimization, providing insights into the behavior of functions, establishing necessary conditions for optimality, and developing efficient algorithms for solving complex problems. By studying these inequalities, researchers can advance our understanding of mathematical and computational sciences and contribute to the development of new technologies and solutions.

Weight: 708g
Dimension: 228 x 152 x 28 (mm)
ISBN-13: 9780128119884