this implies convergence in S ∗ since on compact support it coincides with convergence in D $^{∗}$, but on unbounded support growth at infinity needs to be controlled for convergence in S $^{∗}$. Thus, for any generic estimator ˆ w$_{n}$ represented by a regular function, ˆ w ( z )$_{n}$ , of a regular generalized function w = w ( z ) in S ∗ 1 $_{,m}$( V ) define a corresponding estimator ˜ w$_{n}$ = ˜ w ( z )$_{n}$ by setting it to ˆ w ( z )$_{n}$ if | ˆ w ( z )$_{n}$ | < V (1 + z $^{2}$) $^{m}$, and V (1 + z $^{2}$) m otherwise. Lemma 3. Suppose that the estimator ˆ w$_{n}$ converges to w in probability uniformly on bounded sets; then ˜ w$_{n}$ converges to w in probability in S ∗ and the corresponding Fourier transform ˆ ε$_{n}$ = F t ( ˜ w$_{n}$ ) converges in probability in S ∗ to ε = F t ( w ) . Proof. Consider a set ψ$_{1}$, ..., ψ$_{v}$ ∈ S. For any ζ > 0 find a compact set Λ such that for ˜ ψ representing any of ψ$_{1}$, ..., ψ$_{v}$, F t ( ψ$_{1}$ ) , ..., F t ( ψ$_{v}$ ) Consider | ˜ w$_{n}$ − w | ≤ | ˆ w$_{n}$ − w | I ( z ∈ Λ)+ | ˜ w$_{n}$ − ˆ w$_{n}$ | I ( z ∈ Λ)+ | ˜ w$_{n}$ − w | I ( z ∈ R $^{d}$⧹ Λ ) . Then for any ζ$_{1}$ there exists N such that Pr (∣ ∣ ( ˜ w$_{n}$ − w, ˜ ψ )∣ ∣ > ζ$_{1}$ ) ≤ since the value ∣ ( ˜ w$_{n}$ − w, ˜ ψ )∣ ∣ outside Λ is bonded with certainty by 2 ζ and the two probabilities on bounded Λ can be bounded because of uniform con- vergence in probability of the estimators. Recall that for Fourier transforms ∫$_{R}$ $^{d}$∖ Λ V (1 + t $^{2}$) $^{m}$| ˜ ψ ( t ) | dt < ζ. Pr ( sup Λ | ˆ w$_{n}$ − w | | ∫ ˜ ψ ( t ) dt | > ζ$_{1}$ ) +Pr ( z ∈ Λ : | ˜ w$_{n}$ − ˆ w$_{n}$ | | ∫ ˜ ψ ( t ) dt | > ζ$_{1}$ − 2 ζ ) ≤ ζ$_{1}$, 58