  The approximation of integral type functionals is studied for discrete
observations of a continuous It\^o semimartingale. Based on novel
approximations in the Fourier domain, central limit theorems are proved for
$L^2$-Sobolev functions with fractional smoothness. An explicit $L^2$-lower
bound shows that already lower order quadrature rules, such as the trapezoidal
rule and the classical Riemann estimator, are rate optimal, but only the
trapezoidal rule is efficient, achieving the minimal asymptotic variance.
