content_image/312699.jpg content_image/312700.jpg content_image/312701.jpg content_image/312702.jpg content_image/312697.jpg
The result now follows by Theorem 2.4 of Demko, Moss, and Smith (1984) (which states that ∣ ( A − $^{1}$)$_{i,j}$ ∣ ≤ Cλ | i − j | for all i, j ) and geometric summation. 

Proof of Theorem 5.1. We first prove the univariate case (i.e. d = 1) before generalizing to the multivariate case. 

By the definition of wavelet basis, we may assume without loss of generality that b$_{K}$$_{1}$ = ϕ$_{J,}$$_{0}$ , . . . , b$_{KK}$ = ϕ$_{J,}$$_{2}$ $^{J}$− 1 with K = 2 $^{J}$. 

For any x ∈ [0 , 1] the vector b $^{K}$( x ) has, at most, 2 N elements that are nonzero (as a consequence of the compact support of the ϕ$_{J,k}$ ). It follows that 

<img src='content_image/312699.jpg'>

uniformly in x . Therefore sup$_{x}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ ‖ b $^{K}$( x ) ‖ ℓ 1 ≲ √$_{K}$$_{. Let}$ k be such that N ≤ k ≤ 2 J − N − 1. By boundedness of f$_{X}$ and a change of variables, we have (with µ denoting Lebesgue measure) 

where ‖ ϕ ‖ L $^{1}$( µ ) < ∞ because ϕ has compact support and is continuous. Similar arguments can be used to show the same for the N left and right scaling functions. It follows that max$_{k}$ ‖ b$_{Kk}$ ‖ L $^{1}$( X ) ≲ 2 − J/ 2 = K − 1 / $^{2}$. Therefore, the b$_{K}$$_{1}$, . . . , b$_{KK}$ satisfy the conditions of Lemma 7.1 and hence ‖ P$_{K}$ ‖ ∞ ≲ ‖ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] ) − 1 ‖ ℓ ∞ . It remains to prove that E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] − 1 ℓ ∞ ≲ 1. We first verify the conditions of Lemma 

‖ ( ) ‖ 7.2. Disjoint support of the ϕ$_{J,k}$ implies that ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$])$_{k,j}$ = 0 whenever | k − j | > 2 N − 1. For positive definiteness, we note that 

<img src='content_image/312700.jpg'>

<img src='content_image/312701.jpg'>

<img src='content_image/312702.jpg'>

<img src='content_image/312697.jpg'>

(where we understand the integral performed element wise) because ϕ$_{J,}$$_{0}$, . . . , ϕ$_{J,}$$_{2}$ $^{J}$− 1 are an orthonor- mal basis for V$_{J}$ with respect to the L $^{2}$([0 , 1]) inner product. Similarly, λ$_{min}$ ( E [ b $^{K}$( X ) b $^{K}$( X ) $^{′}$]) ≥ 

