The Maximum Trigonometric Degrees of Quadrature Formulae with Prescribed Nodes

Abstract

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 $m ≥ 3$.