Recall that an integrable function f f on a cube Q 0 {Q_0} in R n {{\mathbf {R}}^n} is said to be of bounded mean oscillation if there is a constant K K such that for every parallel subcube Q Q of Q 0 {Q_0} there exists a constant a Q {a_Q} such that ∫ Q | f − a Q | ≤ K | Q | \int _Q {|f - {a_Q}| \leq K|Q|} , where | Q | |Q| denotes the volume of Q Q . We prove here that if there is an integer d d and a constant K K such that for every parallel subcube Q Q of Q 0 {Q_0} there exists a polynomial p Q {p_Q} of degree ≤ d \leq d such that ∫ Q | f − p Q | ≤ K | Q | \int _Q {|f - {p_Q}| \leq K|Q|} , then f f is of bounded mean oscillation.