A method is presented for linear estimations of functionals of deterministic signals containing additive noise. The method is based on statistical decision theory and assumes discrete observations. In general terms a deterministic signal, with the noise subtracted, is a member of a class of functions with no probability distribution over the members of the class. In this paper the class is restricted to real one-dimensional functions parametrized by a real vector. The linear minimax estimate of the function to be estimated is proposed and the problem of computing it shown to be equivalent to a quadratic programming problem which can be solved exactly when the class of true signals is finite and sometimes when the class is infinite. In the latter case the problem can be solved approximately, subject to some mild restrictions on the signal. The exact algebraic solution is given for prediction of linear signals for up to three observations and is compared with the solution based on Wiener's theory.