Assumptions 1 and 6 are standard regularity conditions used in derivation of optimal uniform con- vergence rates (Stone, 1982; Tsybakov, 2009). Assumption 1(iii) implies Assumption 3. Assumptions 1 and 7 imply Assumption 4 with ζ$_{K,n}$ ≲ √$_{K}$ and λ$_{K,n}$ ≲ 1. Let h ∗ 0 ,K ∈ B$_{K}$ solve inf$_{h}$$_{∈}$$_{B}$$_{K}$ ‖ h$_{0}$ − h ‖ ∞ $_{,w}$. Assumptions 1, 6 and 7 imply that ‖ h$_{0}$ − h ∗ 0 ,K ‖ ∞ ,w ≲ K − p/d (see, e.g. DeVore and Lorentz (1993), Huang (1998), Chen (2007)). Previously Huang (2003b) showed that ‖ P$_{K,w,n}$ ‖ ∞ ,w ≲ 1 wpa1 for spline bases with i.i.d. data. In the proof of Theorem 2.1 we extend his result to allow for weakly dependent regressors. In addition, Theorem 5.2 in Section 5 shows that ‖ P$_{K,w,n}$ ‖ ∞ ,w ≲ 1 wpa1 for wavelet bases with i.i.d. or weakly dependent regressors. Theorem 2.1 Let Assumptions 1, 2(i)(ii)(iii) (with δ ≥ d/p ), 6 and 7 hold. If K ≍ ( n/ log n ) d/ (2 p + d $^{)}$, then provided that either (a), (b), or (c) is satisfied: ‖ ̂ h − h$_{0}$ ‖ ∞ ,w = O$_{p}$ (( n/ log n ) − p/ (2 p + d $^{)}$) (a) { ( X$_{i}$, Y$_{i}$ ) } n i =1 is i.i.d.; (b) { X$_{i}$ } n i =1 is exponentially β -mixing and d < 2 p ; (c) { X$_{i}$ } n i =1 is algebraically β -mixing at rate γ and (2 + γ ) d < 2 γp . Theorem 2.1 states that the optimal uniform convergence rates of Stone (1982) are achieved by spline and wavelet series LS estimators with i.i.d. data whenever δ ≥ d/p . If the regressors are exponen- tially β -mixing the optimal rate of convergence is achieved with δ ≥ d/p and d < 2 p . The restrictions δ ≥ d/p and (2 + γ ) d < 2 γp for algebraically β -mixing (at a rate γ ) reduces naturally towards the exponentially β -mixing restrictions as the dependence becomes weaker (i.e. γ becomes larger). In all cases, for a fixed dimension d ≥ 1, a smoother function (i.e. bigger p ) means a lower value of δ , and hence fatter-tailed error terms ϵ$_{i}$ , are permitted while still obtaining the optimal uniform convergence rate. In particular this is achieved with δ = d/p < 2. Discussion of closely related results . Under Assumption 1 with i.i.d. data and compact X , we can set the weight to be w$_{n}$ = 1 for all n . Let P$_{K}$ denote the L $^{2}$( X ) orthogonal projection operator onto B$_{K}$ , given by P$_{K}$h ( x ) = b $^{K}$( x ) ′ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] ) − $^{1}$E [ b $^{K}$( X$_{i}$ ) h ( X$_{i}$ )] (16) 14