and (ii) max$_{1}$$_{≤}$$_{k}$$_{≤}$$_{K}$ ‖ b$_{Kk}$ ‖ L $^{1}$( X ) ≲ c − 1 K , then Proof of Lemma 7.1. By H¨ older’s inequality (with (i)), definition of the operator norm, and H¨ older’s inequality again (with (ii)), we obtain: ‖ P$_{K}$ ‖ ∞ ≲ ‖ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] ) − 1 ‖ ℓ ∞ . | P$_{K}$f ( x ) | = | b $^{K}$( x ) ′ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] ) − $^{1}$E [ b $^{K}$( X$_{i}$ ) f ( X$_{i}$ )] | ≤ ‖ b $^{K}$( x ) ‖ ℓ 1 ‖ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] ) − $^{1}$E [ b $^{K}$( X$_{i}$ ) f ( X$_{i}$ )] |‖ ℓ ∞ ≲ c$_{K}$ ‖ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] ) − $^{1}$E [ b $^{K}$( X$_{i}$ ) f ( X$_{i}$ )] |‖ ℓ ∞ ≤ c$_{K}$ ‖ E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] − 1 ‖ ℓ ∞ ‖ E [ b $^{K}$( X$_{i}$ ) f ( X$_{i}$ )] |‖ ℓ ∞ = c$_{K}$ ‖ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] ) − 1 ‖ ℓ ∞ max 1 ≤ k ≤ $_{K}$E [ | b$_{Kk}$ ( X$_{i}$ ) f ( X$_{i}$ ) | ] ≤ c$_{K}$ ‖ E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] − 1 ‖ ℓ ∞ max 1 ≤ k ≤ $_{K}$E [ | b$_{Kk}$ ( X$_{i}$ ) | ] ‖ f ‖ ∞ ≲ ‖ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] ) − 1 ‖ ∞ ‖ f ‖ ∞ uniformly in x . The result now follows by taking the supremum over x ∈ X . We will bound ‖ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] ) − 1 ‖ ℓ ∞ for (tensor product) wavelet bases using the following Lemma. Lemma 7.2 Let A ∈ R K × K be a positive definite symmetric matrix such that A$_{i,j}$ = 0 whenever | i − j | > m/ 2 for m even. Then: ‖ A − $^{1}$‖ ℓ ∞ ≤ 2 C 1 − λ where κ = λ$_{max}$ ( A ) /λ$_{min}$ ( A ) λ = ( √$_{κ}$ − 1 √$_{κ}$$_{+ 1}$ ) 2 /m < 1 C = ‖ A − $^{1}$‖ max { 1 , (1 + √$_{κ}$$_{)}$ $^{2}$/ (2 κ ) } . ‖ A − $^{1}$‖ ∞ = max j ≤ K K ∑ k $_{=1}$| ( A − $^{1}$)$_{j,k}$ | . Proof of Lemma 7.2. By definition of the matrix infinity norm, we have 47