  We study estimation of (semi-)inner products between two nonparametric
probability distributions, given IID samples from each distribution. These
products include relatively well-studied classical $\mathcal{L}^2$ and Sobolev
inner products, as well as those induced by translation-invariant reproducing
kernels, for which we believe our results are the first. We first propose
estimators for these quantities, and the induced (semi)norms and
(pseudo)metrics. We then prove non-asymptotic upper bounds on their mean
squared error, in terms of weights both of the inner product and of the two
distributions, in the Fourier basis. Finally, we prove minimax lower bounds
that imply rate-optimality of the proposed estimators over Fourier ellipsoids.
