Reproducing kernel Hilbert spaces and Mercer theorem
  We characterize the reproducing kernel Hilbert spaces whose elements are $p$-integrable functions in terms of the boundedness of the integral operator whose kernel is the reproducing kernel. Moreover, for $p=2$ we show that the spectral decomposition of this integral operator gives a complete description of the reproducing kernel. 
Claudio Carmeli, Ernesto De Vito, Alessandro Toigo