Suppose that on the Interval [a, b] the nodes $$a = x_0 0 have continuous (n−i)-th derivatives (i=0, 1, ..., n). Sn,m will designate the subspace of functions that have continuous (n−1)-st derivatives on [a, b] and coincide on each of the intervals [xj, xj+1] (j=0, 1, ..., m) with some polynomial from the system {ui(t)} i=0 n .THEOREM. For every continuous function on [a, b] there exists in Sn,m a unique element of best mean approximation.