Let $s$ and $d$ be non-negative integers with $0\leq s<d$ and let $v,u\in \mathbb{R}^d$ be vectors satisfying the sparsity estimate $$ \max\{\|u\|_0,\|v\|_0\}\leq d-s, $$ where, as ususal, for any vector $x \in \mathbb{R}^d$ we define $\|x\|_0:=\sum_{i=1}^d\,I_{x_i\neq 0}$.

Let $A$ be an $d\times d$-matrix solving $Au=v$. How sparse can $A$ be? I.e.: what is $$ \inf_{A\in L(\mathbb{R}^d),\,Au=v}\, \|A\|_0, $$ where as above $\|A\|_0:=\sum_{i,j=1}^d\,I_{A_{i,j}\neq 0}$.

Let $s$ and $d$ be non-negative integers with $0\leq s<d$ and let $v,u\in \mathbb{R}^d$ be vectors satisfying the sparsity estimate $$ \max\{\|u\|_0,\|v\|_0\}\leq d-s, $$ where, as usual, for any vector $x \in \mathbb{R}^d$ we define $\|x\|_0:=\sum_{i=1}^d\,I_{x_i\neq 0}$.

Let $A$ be an $d\times d$-matrix solving $Au=v$. How sparse can $A$ be? I.e.: what is $$ \inf_{A\in L(\mathbb{R}^d),\,Au=v}\, \|A\|_0, $$ where as above $\|A\|_0:=\sum_{i,j=1}^d\,I_{A_{i,j}\neq 0}$.

Let $s$ and $d$ be non-negative integers with $0\leq s<d$ and let $v,u\in \mathbb{R}^d$ be vectors satisfying the sparsity lower-bound $$ d-s \leq \min\{\|u\|_0,\|v\|_0\}, $$ where, as ususal, for any vector $x \in \mathbb{R}^d$ we define $\|x\|_0:=\sum_{i=1}^d\,I_{x_i\neq 0}$.

Let $A$ be an $d\times d$-matrix solving $Au=v$. How sparse can $A$ be? I.e.: what is $$ \inf_{A\in L(\mathbb{R}^d),\,Au=v}\, \|A\|_0, $$ where as above $\|A\|_0:=\sum_{i,j=1}^d\,I_{A_{i,j}\neq 0}$.

Let $s$ and $d$ be non-negative integers with $0\leq s<d$ and let $v,u\in \mathbb{R}^d$ be vectors satisfying the sparsity estimate $$ \max\{\|u\|_0,\|v\|_0\}\leq d-s, $$ where, as ususal, for any vector $x \in \mathbb{R}^d$ we define $\|x\|_0:=\sum_{i=1}^d\,I_{x_i\neq 0}$.

Let $A$ be an $d\times d$-matrix solving $Au=v$. How sparse can $A$ be? I.e.: what is $$ \inf_{A\in L(\mathbb{R}^d),\,Au=v}\, \|A\|_0, $$ where as above $\|A\|_0:=\sum_{i,j=1}^d\,I_{A_{i,j}\neq 0}$.