Quadrature formulae with multiple nodes and a maximal trigonometric degree of exactness
The authors construct for the integral \(\int_{-\pi}^{\pi} w(x)f(x)dx\) an interpolatory quadrature formulae with multiple nodes, which have the maximal trigonometric degree of exactness. For that purpose they introduce and consider the so-called \(\sigma\)-orthogonal trigonometric polynomials of semi-integer degree and give a numerical method for their construction. Some numerical examples are given. For the weight function \(w(x)=1\) a thus study was considered by \textit{D. P. Dryanov} [Numer. Math. 67, No. 4, 441--464 (1994; Zbl 0798.41023)].
trigonometric polynomials, inequalities, extremal problems
scientific article; zbMATH DE number 5571605
scientific article