We study expected Riesz s-energies and linear statistics of some determinantal processes on the sphere S d . In particular, we compute the expected Riesz and logarithmic energies of the determinantal processes given by the reproducing kernel of the space of spherical harmonics. This kernel defines the so called harmonic ensemble on S d . With these computations we improve previous estimates for the discrete minimal energy of configurations of points in the sphere. We prove a comparison result for Riesz 2-energies of points defined through determinantal point processes associated to isotropic kernels. As a corollary we get that the Riesz 2-energy of the harmonic ensemble is optimal among ensembles defined by isotropic kernels with the same trace. Finally, we study the variance of smooth and rough linear statistics for the harmonic ensemble and compare the results with the variance for the spherical ensemble (in S 2 ).

ABSTRACT. We prove upper pointwise estimates for the Bergman kernel of the weighted Fock space of entire functions in L 2 (e −2φ ) where φ is a subharmonic function with ∆φ a doubling measure. We derive estimates for the canonical solution operator to the inhomogeneous CauchyRiemann equation and we characterize the compactness of this operator in terms of ∆φ.

We find necessary density conditions for Marcinkiewicz-Zygmund inequalities and interpolation for spaces of spherical harmonics in S d with respect to the L p norm. Moreover, we prove that there are no complete interpolation families for p = 2.

ABSTRACT. The Fekete points are the points that maximize a Vandermonde-type determinant that appears in the polynomial Lagrange interpolation formula. They are well suited points for interpolation formulas and numerical integration. We prove the asymptotic equidistribution of the Fekete points in the sphere. The way we proceed is by showing their connection with other array of points, the Marcinkiewicz-Zygmund arrays and the interpolating arrays, that have been studied recently.