content_image/321874.jpg content_image/321875.jpg content_image/321879.jpg content_image/321880.jpg
and (ii) max$_{1}$$_{≤}$$_{k}$$_{≤}$$_{K}$ ‖ b$_{Kk}$ ‖ L $^{1}$( X ) ≲ c − 1 K , then 

<img src='content_image/321874.jpg'>

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: 

uniformly in x . The result now follows by taking the supremum over x ∈ X . − 1 

We will bound ‖ ( E [ b $^{K}$( X$_{i}$ ) b $^{K}$( X$_{i}$ ) $^{′}$] ) ‖ ℓ ∞ for (tensor product) wavelet bases using the following Lemma. 

<img src='content_image/321875.jpg'>

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 

<img src='content_image/321879.jpg'>

Proof of Lemma 7.2. By definition of the matrix infinity norm, we have 

<img src='content_image/321880.jpg'>

