content_image/295654.jpg content_image/295659.jpg content_image/295660.jpg
We now present conditions under which the empirical projection onto a tensor-product wavelet basis is stable wpa1. Here the projection operator is 

<img src='content_image/295654.jpg'>

where the elements of b K span the tensor products of d univariate spaces Wav( K$_{0}$, [0 , 1]). The following Theorem states simple sufficient conditions for ‖ P$_{K,n}$ ‖ ∞ ≲ 1 wpa1. 

Theorem 5.2 Let conditions stated in Theorem 5.1 hold. Then ‖ P$_{K,n}$ ‖ ∞ ≲ 1 wpa1 provided that either (a), (b), or (c) is satisfied: 

(a) { X$_{i}$ } n i =1 are i.i.d. and √ ( K log n ) /n = o (1) n 

(b) { X$_{i}$ } i =1 are exponentially β -mixing and √ K (log n ) $^{2}$/n = o (1) , or n γ/ 

(c) { X$_{i}$ } i =1 are algebraically β -mixing at rate γ and √ ( K log n ) /n (1+ γ ) = o (1) . 

## 6 Brief review of B-spline and wavelet sieve spaces 

We first outline univariate B-spline and wavelet sieve spaces on [0 , 1], then deal with the multivariate case by constructing a tensor-product sieve basis. 

B-splines B-splines are defined by their order r ≥ 1 (or degree r − 1 ≥ 0) and number of interior knots m ≥ 0. Define the knot set 

<img src='content_image/295659.jpg'>

We generate a L $^{∞}$-normalized B-spline basis recursively using the De Boor relation (see, e.g., Chapter 5 of DeVore and Lorentz (1993)) then appropriately rescale the basis functions. Define the interior intervals I$_{1}$ = [ t$_{0}$, t$_{1}$ ) , . . . , I$_{m}$ = [ t$_{m}$, t$_{m}$$_{+1}$ ] and generate a basis of order 1 by setting 

<img src='content_image/295660.jpg'>

