  We consider the symmetric Toeplitz matrix completion problem, whose matrix
under consideration possesses specific row and column structures. This problem,
which has wide application in diverse areas, is well-known to be
computationally NP-hard. This note provides an upper bound on the objective of
minimizing the rank of the symmetric Toeplitz matrix in the completion problem
based on the theorems from the trigonometric moment problem and semi-infinite
problem. We prove that this upper bound is less than twice the number of linear
constraints of the Toeplitz matrix completion problem. Compared with previous
work in the literature, ours is one of the first efforts to investigate the
bound of the objective value of the Toeplitz matrix completion problem.
