and Lemma 3, and the third by convergence of ˆ ε$_{1}$$_{n}$. Then ˜ γ$_{n}$ converges in probability to γ = φ − $^{1}$ε$_{1}$ in S $^{∗}$. Taking inverse Fourier transforms in S ∗ concludes the proof. The Theorem provides consistency of plug-in estimators for solutions to the system of equations (16) and consequently (3) in a fairly general set-up. Nevertheless some assumptions can be further relaxed. Of course, establish- ing results for a compact support of the Fourier transforms is much easier and thus using spectral cut-off can be advantageous especially when high frequency components of the regression function may be commesurate with the magnitude of the error components. Computation of the estimators requires applying Fourier transforms and inverse Fourier transforms. This can be accomplished with numerical algo- rithms. However, it is possible to simplify the estimated ˆ ε . Consider instead of the estimator in (22) , ˆ w$_{1}$ ( z )$_{n}$ , an estimator computed as n ∑ i =1 α − 1 i y$_{i}$K z$_{i}$ − z h ) with the weight α$_{i}$ = n ∑ j = i K ( z$_{j}$ − z$_{i}$ h ) replacing the z − dependent weight in ˆ w$_{1}$ ( z )$_{n}$ and similar estimators for ˆ w$_{2}$$_{k}$ ( z )$_{n}$ . Then the corresponding Fourier transform ˆ ε$_{1}$$_{n}$ ( s ) can be expressed as n ∑ i =1 α − 1 i y$_{i}$e is $^{$^{T}$z$_{j}$}$sinc ( s $^{T}$z$_{j}$ π ) , where by def- inition sinc ( x ) = sin πx πx , and similar expressions for ˆ ε$_{2}$$_{kn}$. Further computation for the estimators would have to be done numerically. 67