On the Hilbert kernel density estimate

Let X be an -valued random variable with unknown density f. Let X1,...,Xn be i.i.d. random variables drawn from f. We study the pointwise convergence of a new class of density estimates, of which the most striking member is the Hilbert kernel estimatewhere Vd is the volume of the unit ball in . This is particularly interesting as this density estimate is basically of the format of the kernel estimate (except for the log n factor in front) and the kernel estimate does not have a smoothing parameter.