  We classify the quasifinite highest weight modules over a family of
subalgebras W_{\infty}^{n} of the central extension W_{1+\infty} of the Lie
algebra of differential operators on the circle consisting of operators of
order \geq n. We classify the unitary quasifinite highest weight modules over
W_{\infty}=W_{\infty}^{1} and realize them in terms of unitary highest weight
representations of the Lie algebra of infinite matrices with finitely many
non-zero diagonals.
