where b$_{K}$$_{1}$, . . . , b$_{KK}$ are a collection of K sieve basis functions and Second, we consider convergence in the (sequence of) weighted sup norm(s) ‖ · ‖ ∞ ,w given by This modification is made because simple functions, such as polynomials of x , have infinite sup norm when X$_{i}$ has unbounded support, but will have finite weighted sup norm. Remark 2.1 When X is compact we may simply set w$_{n}$ ( x ) = 1 for all x ∈ X and all n . With such a choice of weighting, the series LS estimator with weighted basis trivially coincides with the series LS estimator in (2) with unweighted basis, and ‖ · ‖ ∞ ,w = ‖ · ‖ $_{∞}$. When X is unbounded there are several possible choices for D $_{n}$. For instance, we may take D n = D for all n , where D ⊆ X is a fixed compact convex set with nonempty interior. This approach is not without precedent in the nonparametric analysis of nonlinear time series models. For example, Huang and Shen (2004) use a similar approach to trim extreme observations in nonparametric func- tional coefficient regression models, following Tjøstheim and Auestad (1994). More generally, we can consider an expanding sequence of compact nonempty sets D n ⊂ X with D n ⊆ D n +1 for all n and set w$_{n}$ ( x ) = { x ∈ D n } for all n . For example, if X = R d we could take D n = { x ∈ R d : ‖ x ‖ p ≤ r$_{n}$ } where 0 < r$_{n}$ ≤ r$_{n}$$_{+1}$ < ∞ for all n . This approach is similar to excluding functions far from the support of the data when performing series LS estimation with a compactly-supported wavelet basis for L $^{2}$( $^{R}$) or L $^{2}$( R $^{d}$). We defer estimation with smooth weighting functions of the form w$_{n}$ ( x ) = (1 + ‖ x ‖ $^{2}$) − ω or w$_{n}$ ( x ) = exp( −‖ x ‖ $^{ω}$) to future research. We first introduce some mild regularity conditions that are satisfied by typical regression models and most linear sieve bases. Assumption 1 (i) { X$_{i}$ } ∞ i = −∞ is strictly stationary, (ii) X ⊆ R d is convex and has nonempty interior. 8 b K $_{w}$( x ) = ( b$_{K}$$_{1}$ ( x ) w$_{n}$ ( x ) , . . . , b$_{KK}$ ( x ) w$_{n}$ ( x )) ′ (8) B$_{w}$ = ( b K $_{w}$( X$_{1}$ ) , . . . , b K $_{w}$( X$_{n}$ )) $^{′}$. (9) ‖ f ‖ ∞ ,w = sup x | f ( x ) w$_{n}$ ( x ) | = sup x ∈D$_{n}$ | f ( x ) | (10)