The Maximum Trigonometric Degrees of Quadrature Formulae with Prescribed Nodes (Communications in Mathematical Research, 2021)
The purpose of this paper is to study the maximum trigonometric degree of the quadrature formula associated with $m$ prescribed nodes and $n$ unknownadditional nodes in the interval $(−π, π]$. We show that for a fixed $n$, the quadratureformulae with $m$ and $m + 1$ prescribed nodes share the same maximum degree if $m$ isodd. We also give necessary and sufficient conditions for all the additional nodes to bereal, pairwise distinct and in the interval $(−π, π]$ for even $m$, which can be obtainedconstructively. Some numerical examples are given by choosing the prescribed nodesto be the zeros of Chebyshev polynomials of the second kind or randomly for $m ≥ 3$.