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I'm trying to read Donoho's 2004 paper Compressed Sensing and am having trouble with a supposedly trivial statement (equation 1.2 on page 3).

He makes the sparsity assumption on $\theta \in \mathbb{R}^m$ that for some $0<p<2$ and $R>0$ we have $\|\theta\|_p\leq R$. Then if $\theta_N$ denotes $\theta$ with everything except the $N$ largest coefficients set to $0$ he claims that $\| \theta-\theta_N \|_2 \leq \zeta_{2,p} \cdot \| \theta \|_p \cdot (N+1)^{1/2-1/p}$ for $N=0,1,2,\ldots$ where $\zeta_{2,p}$ depends only on $p$.

I've trying tried writing out the definitions of various things. I've noticed that the $N$th largest coefficient must satisfy $\mid\theta_i\mid \leq RN^{-1/p}$ but I can't figure out how the result above follows.

I'm also having some difficulty thinking about $\ell^p$ spaces with $0<p<1$, in particular knowing what results from the $p>1$ theory apply. Does anyone know some good notes or a book that covers this?

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# l^p space inequality related to compressed sensing

I'm trying to read Donoho's 2004 paper Compressed Sensing and am having trouble with a supposedly trivial statement (equation 1.2 on page 3).

He makes the sparsity assumption on $\theta \in \mathbb{R}^m$ that for some $0<p<2$ and $R>0$ we have $\|\theta\|_p\leq R$. Then if $\theta_N$ denotes $\theta$ with everything except the $N$ largest coefficients set to $0$ he claims that $\| \theta-\theta_N \|_2 \leq \zeta_{2,p} \cdot \| \theta \|_p \cdot (N+1)^{1/2-1/p}$ for $N=0,1,2,\ldots$ where $\zeta_{2,p}$ depends only on $p$.

I've trying writing out the definitions of various things. I've noticed that the $N$th largest coefficient must satisfy $\mid\theta_i\mid \leq RN^{-1/p}$ but I can't figure out how the result above follows.

I'm also having some difficulty thinking about $\ell^p$ spaces with $0<p<1$, in particular knowing what results from the $p>1$ theory apply. Does anyone know some good notes or a book that covers this?