(a) { ( X$_{i}$, Y$_{i}$ ) } n i =1 is i.i.d.; (b) { X$_{i}$ } n i =1 is weakly dependent: Assumption 8 holds, and ‖ ˜ B ′ w ˜ B$_{w}$/n − I$_{K}$ ‖ = o$_{p}$ ( K − 1 / $^{2}$) . We now consider the special case of irregular but sup-norm bounded linear or nonlinear functionals as discussed in Remark 3.1. The sup-norm convergence rates for series LS estimators in Section 2 are employed to derive asymptotic normality of plug-in estimators of such functionals under weak conditions. To save space, for the weakly dependent case we only present sufficient conditions for asymptotic normality of f ( ̂ h ) when the regression error has no more than a finite 4th absolute moment (i.e., E [ | ϵ$_{i}$ | 2+ $^{δ}$] < ∞ for some 0 ≤ δ ≤ 2). We also take X = [0 , 1] d and w$_{n}$ = 1 for all n for simplicity. Corollary 3.1 Let f be an irregular but sup-norm bounded linear functional, and let Assumptions 1 (with X = [0 , 1] $^{d}$), 2(i)(ii)(iv)(v), 6, and 7 hold. Then as n, K → ∞ provided that either (a), (b) or (c) is satisfied: (a) { ( X$_{i}$, Y$_{i}$ ) } n i =1 is i.i.d.: √$_{nV}$ − 1 / 2 K K − p/d = o (1) and ( K log K ) /n = o (1) ; (b) { ( X$_{i}$, Y$_{i}$ ) } n i =1 is exponentially β -mixing: Assumption 2(iii) also holds, √$_{nV}$ − 1 / 2 K K − p/d = o (1) , and K (2+ δ ) $^{/δ}$(log n ) $^{2}$/n = o (1) with δ ≤ 2 ; (c) { ( X$_{i}$, Y$_{i}$ ) } n i =1 is algebraically β -mixing at rate γ : Assumption 2(iii) also holds, √$_{nV}$ − 1 / 2 K K − p/d = o (1) , and K (2+ δ ) $^{/δ}$(log K ) /n γ/ (1+ γ ) = o (1) with δ ≤ 2 . Corollary 3.1 part (a) extends the weakest known result on pointwise asymptotic normality of spline LS estimators in Huang (2003b) to general sup-norm bounded linear functionals of spline or wavelet series LS estimators. 9 Corollary 3.2 Let f be an irregular but sup-norm bounded nonlinear functional, and let Assumptions 1 (with X = [0 , 1] $^{d}$), 2, 6, 7 and 9(i’)(ii’)(iii’) hold. Then $^{√$_{n}$$_{(}$$_{f}$$_{(}$}$̂ h ) − f ( h$_{0}$ )) V 1 / 2 K → d N (0 , 1) $^{9}$Under the assumption of empirical identifiability (see equation (30)) and other conditions similar to the ones listed in Corollary 3.1 part (a), Chen and Huang (2003) derived the asymptotic normality of plug-in spline LS estimators of sup-norm bounded linear functionals (see their Theorem 4). 22 $^{√$_{n}$$_{(}$$_{f}$$_{(}$}$̂ h ) − f ( h$_{0}$ )) V 1 / 2 K → d N (0 , 1)