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,}$$_{2}$ $^{J}$− 1 with K = 2 $^{J}$. For any x ∈ [0 , 1] the vector b $^{K}$( x ) has, at most, 2 N elements that are nonzero (as a consequence of the compact support of the ϕ$_{J,k}$ ). It follows that uniformly in x . Therefore sup$_{x}$$_{∈}$$_{[0}$$_{,}$$_{1]}$ ‖ b $^{K}$( x ) ‖ ℓ 1 ≲ √$_{K}$$_{. 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}$$_{1}$, . . . , b$_{KK}$ satisfy the conditions of Lemma 7.1 and hence ‖ P$_{K}$ ‖ ∞ ≲ ‖ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] ) − 1 ‖ ℓ ∞ . − 1 ∞ It remains to prove that ‖ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] ) − 1 ‖ ℓ ∞ ≲ 1. We first verify the conditions of Lemma 7.2. Disjoint support of the ϕ$_{J,k}$ implies that ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$])$_{k,j}$ = 0 whenever | k − j | > 2 N − 1. For positive definiteness, we note that (where we understand the integral performed element wise) because ϕ$_{J,}$$_{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 ) $^{′}$]) ≥ λ$_{max}$ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$]) ≤ ( sup x ∈ [0 , 1] f$_{X}$ ( x ) λ$_{max}$ ( ∫ [0 , 1] b $^{K}$( x ) b $^{K}$( x ) $^{′}$d µ ( x ) = ( sup x ∈ [0 , 1] f$_{X}$ ( x ) (128) 48 ‖ b $^{K}$( x ) ‖ ℓ 1 ≤ (2 N )2 J/ $^{2}$max {‖ ϕ ‖ $_{∞}$, ‖ ϕ l 0 ‖ $_{∞}$, . . . , ‖ ϕ l N − 1 ‖ $_{∞}$, ‖ ϕ r − 1 ‖ $_{∞}$, . . . , ‖ ϕ r − N ‖ ∞ } ≲ 2 J/ 2 (124) E [ | ϕ$_{J,k}$ ( X$_{i}$ ) | ] ≤ sup x ∈ [0 , 1] f$_{X}$ ( x ) ∫ R 2 J/ $^{2}$| ϕ (2 $^{J}$x − k ) | d µ ( x ) (125) = sup x ∈ [0 , 1] f$_{X}$ ( x )2 − J/ 2 ∫ R | ϕ ( y ) | d µ ( y ) (126) = sup x ∈ [0 , 1] f$_{X}$ ( x )2 − J/ $^{2}$‖ ϕ ‖ L $^{1}$( µ ) (127)