
NuHAG :: TALKS
Talks given at NuHAG events


Carl's inequality: Optimality and Lower Bounds in Approximation Theory Ian Vibyral given at NuHAG seminar (21.06.17 11:30) id: 3326 length: 40min status: type: LINKPresentation: ABSTRACT:
TeX: Classical Carl's inequality states that for any two Banach spaces $X$ and $Y$ and any $\alpha>0$ there exists a constant $\gamma_\alpha>0$ such that $$ \sup_{1\le k\le n}k^{\alpha}e_k(T)\le \gamma_\alpha \sup_{1\le k\le n} k^\alpha c_k(T) $$ holds for all linear and bounded operators $T:X\to Y$. Here $e_k(T)$ is the $k$th entropy number of $T$ and $c_k(T)$ is the $k$th Gelfand number of $T$. This inequality has been used to obtain lower bounds on Gelfand numbers, which in turn are a useful tool in the study of optimality of algorithms. We extend the Carl's inequality to quasiBanach spaces, which allows to extend this approach also to the area of sparse recovery, where $\ell_p^N$ spaces play an important role also for $p
