Commun. Math. Res., 30 (2014), pp. 334-344.
Published online: 2021-05
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The purpose of this paper is to study the maximum trigonometric degree of the quadrature formula associated with $m$ prescribed nodes and $n$ unknown additional nodes in the interval $(−π, π]$. We show that for a fixed $n$, the quadrature formulae with $m$ and $m + 1$ prescribed nodes share the same maximum degree if $m$ is odd. We also give necessary and sufficient conditions for all the additional nodes to be real, pairwise distinct and in the interval $(−π, π]$ for even $m$, which can be obtained constructively. Some numerical examples are given by choosing the prescribed nodes to be the zeros of Chebyshev polynomials of the second kind or randomly for $m ≥ 3$.
}, issn = {2707-8523}, doi = {https://doi.org/10.13447/j.1674-5647.2014.04.07}, url = {http://global-sci.org/intro/article_detail/cmr/18957.html} }The purpose of this paper is to study the maximum trigonometric degree of the quadrature formula associated with $m$ prescribed nodes and $n$ unknown additional nodes in the interval $(−π, π]$. We show that for a fixed $n$, the quadrature formulae with $m$ and $m + 1$ prescribed nodes share the same maximum degree if $m$ is odd. We also give necessary and sufficient conditions for all the additional nodes to be real, pairwise distinct and in the interval $(−π, π]$ for even $m$, which can be obtained constructively. Some numerical examples are given by choosing the prescribed nodes to be the zeros of Chebyshev polynomials of the second kind or randomly for $m ≥ 3$.