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

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 quasi-Banach 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 $p < 1$. ...

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