TY - JOUR
T1 - The Maximum Trigonometric Degrees of Quadrature Formulae with Prescribed Nodes
AU - Luo , Zhongxuan
AU - Yu , Ran
AU - Meng , Zhaoliang
JO - Communications in Mathematical Research 
VL - 4
SP - 334
EP - 344
PY - 2021
DA - 2021/05
SN - 30
DO - http://doi.org/10.13447/j.1674-5647.2014.04.07
UR - https://global-sci.org/intro/article_detail/cmr/18957.html
KW - quadrature formula, trigonometric function, bi-orthogonality, truncated
complex moment problem.
AB - <p style="text-align: justify;">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 $m ≥ 3$.</p>