• diagonalization of infinite-dimensional parameters;
  • projection estimators;
  • spatial autoregressive functional series;
  • spatial functional prediction;
  • two-parameter diffusion processes

The class of spatial autoregressive Hilbertian models (SARH(1) processes) is considered. The projection estimation methodology proposed here is based on the biorthogonal eigenfunction bases diagonalizing the infinite-dimensional parameters involved in the SARH(1) state equation. These bases remove the ill-posed nature of the functional equation system defining the moment-based estimators of such parameters. The performance of the proposed projection estimation methodology, in the SARH(1) context, is illustrated in terms of simulated and real-data examples. In particular, this methodology provides a suitable spatial functional extrapolation of tropical and subtropical weak-dependence ocean surface temperature profiles, in the absence of high spatial concentration of weather stations, removing computational problems associated with matrix determinant close to zero. Copyright © 2011 John Wiley & Sons, Ltd.