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We consider the estimation of the value of a linear functional of the slopeparameter in functional linear regression, where scalar responses are modeledin dependence of random functions. The theory in this paper covers inparticular point-wise estimation as well as the estimation of weighted averagesof the slope parameter. We propose a plug-in estimator which is based on adimension reduction technique and additional thresholding. It is shown thatthis estimator is consistent under mild assumptions. We derive a lower boundfor the maximal mean squared error of any estimator over a certain ellipsoid ofslope parameters and a certain class of covariance operators associated withthe regressor. It is shown that the proposed estimator attains this lower boundup to a constant and hence it is minimax optimal. Our results are appropriateto discuss a wide range of possible regressors, slope parameters andfunctionals. They are illustrated by considering the point-wise estimation ofthe slope parameter or its derivatives and its average value over a giveninterval.

On rate optimal local estimation in functional linear model