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.