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We consider the problem of determining the asymptotic order of the Gelfandnumbers of mixed-(quasi-)norm embeddings $\ell^b_p(\ell^d_q) \hookrightarrow\ell^b_r(\ell^d_u)$ given that $p \leq r$ and $q \leq u$, with emphasis oncases with $p\leq 1$ and/or $q\leq 1$. These cases turn out to be related tostructured sparsity. We obtain sharp bounds in a number of interestingparameter constellations. Our new matching bounds for the Gelfand numbers ofthe embeddings of $\ell_1^b(\ell_2^d)$ and $\ell_2^b(\ell_1^d)$ into$\ell_2^b(\ell_2^d)$ imply optimality assertions for the recovery ofblock-sparse and sparse-in-levels vectors, respectively. In addition, we applythe sharp estimates for $\ell^b_p(\ell^d_q)$-spaces to obtain new two-sidedestimates for the Gelfand numbers of multivariate Besov space embeddings inregimes of small mixed smoothness. It turns out that in some particular casesthese estimates show the same asymptotic behaviour as in the univariatesituation. In the remaining cases they differ at most by a $\log\log$ factorfrom the univariate bound.

Gelfand numbers related to structured sparsity and Besov space embeddings with small mixed smoothness