  We show that there are exactly two anti-involution $\sigma_{\pm}$ of the
algebra of differential operators on the circle that are a multiple of
$p(t\partial_t)$ preserving the principal gradation ($p\in\CC[x]$
non-constant). We classify the irreducible quasifinite highest weight
representations of the central extension $\hat{\D}_p^{\pm}$ of the Lie
subalgebra fixed by $-\sigma_{\pm}$. The most important cases are the
subalgebras $\hat{\D}_x^{\pm}$ of $W_{\infty}$, that are obtained when
$p(x)=x$. In these cases we realize the irreducible quasifinite highest weight
modules in terms of highest weight representation of the central extension of
the Lie algebra of infinite matrices with finitely many non-zero diagonals over
the algebra $\CC[u]/(u^{m+1})$ and its classical Lie subalgebras of $C$ and $D$
types.
