Remark on a Problem of Rational Approximation

We show that for any nonincreasing number sequence $\{a_n\}^{\infty}_{n=0}$ converging to zero, there exists a continuous $2\pi$-periodic function $g$ such that the sequence of its best uniform trigonometric rational approximations $\{R_n(g,C_{2\pi})\}^{\infty}_{n=0}$ and the sequence $\{a_n\}^{\infty}_{n=0}$ have the same order of decay.