An Optimal Quadrature Formula with Derivative in the Hilbert Space

Abdullo R. Hayotov () and Samandar S. Babaev ()
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Abdullo R. Hayotov: Uzbekistan Academy of Sciences
Samandar S. Babaev: Uzbekistan Academy of Sciences

Chapter Chapter 9 in Analysis, Approximation, Optimization: Computation and Applications, 2025, pp 155-170 from Springer

Abstract: Abstract This chapter addresses the derivation and analysis of an optimal quadrature formula in the Hilbert space W 2 ( 2 , 1 ) ( t , 1 ) $$W_2^{(2,1)}(t,1)$$ , where functions φ $$\varphi $$ with prescribed properties reside. The quadrature formula is expressed as a linear combination of function values and their first-order derivatives at equidistant nodes in the interval [ t , 1 ] $$[t,1]$$ . The coefficients are determined by minimizing the norm of the error functional in the dual space W 2 ( 2 , 1 ) ∗ ( t , 1 ) $$W_2^{(2,1)*}(t,1)$$ . The error functional is defined as the difference between the integral of a function over the interval and the quadrature approximation. Key results include explicit expressions for the coefficients and the norm of the error functional. The optimization problem is formulated and solved, leading to a system of linear equations for the coefficients. Analytical solutions of the system are obtained, providing an explicit expression for the optimal coefficients. This chapter demonstrates the application of these results to estimate the error of the quadrature formula on functions in W 2 ( 2 , 1 ) ( t , 1 ) $$W_2^{(2,1)}(t,1)$$ .

Date: 2025
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DOI: 10.1007/978-3-031-85743-0_9

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