Construction of optimal quadrature formulas for Fourier coefficients in Sobolev space \(L_{2}^{(m)}(0,1)\)

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• 作者：N. D. Boltaev ; A. R. Hayotov ; Kh. M. Shadimetov
• 关键词：Fourier coefficients ; Optimal quadrature formulas ; The error functional ; Extremal function ; Hilbert space ; Optimal coefficients
• 刊名：Numerical Algorithms
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：74
• 期：2
• 页码：307-336
• 全文大小：
• 刊物类别：Computer Science
• 刊物主题：Numeric Computing; Algorithms; Algebra; Theory of Computation; Numerical Analysis;
• 出版者：Springer US
• ISSN：1572-9265
• 卷排序：74

文摘

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the \(L_{2}^{(m)}(0,1)\) space for numerical calculation of Fourier coefficients. Using the S.L.Sobolev’s method, we obtain new optimal quadrature formulas of such type for N+1≥m, where N+1 is the number of nodes. Moreover, explicit formulas for the optimal coefficients are obtained. We study the order of convergence of the optimal formula for the case m=1. The obtained optimal quadrature formulas in the \(L_{2}^{(m)}(0,1)\) space are exact for P_m−1(x), where P_m−1(x) is a polynomial of degree m−1. Furthermore, we present some numerical results, which confirm the obtained theoretical results.