 Parametric-rate kernel estimation of a bandlimited density

Let a bounded probability density function have Fourier transform supported in the plateau of a flat-top kernel for density estimation. At fixed smoothing bandwidth one, the convolution theorem gives $K\ast f=f$, so the kernel density estimator is pointwise unbiased. Its variance is at most $\Vert f{\Vert }_{\infty }\Vert K{\Vert }_2^2/n$, and the Cauchy-Schwarz inequality proves the displayed rate. An explicit example is $f(x)=(1-\cos x)/(\pi x^2)$, with value $1/(2\pi )$ at zero and transform $(1-|t|{)}_{+}$. Direct inverse Fourier integration gives the formula. The function is nonnegative, integrable and has integral one because its transform at zero is one. Compact frequency support also gives bounded derivatives of every order.