Nonlinear approximation with dictionaries. II: Inverse estimates (Q853542)

A countable family \(D=\{g_k\| \) of unit vectors in a Banach space \(X\) is called a dictionary if it spans a dense subspace of \(X\). For a given \(f\in X\) and a natural number \(m\), let \(\sigma_m\) be the error of the best \(m\)-term approximation by \(D\), that is, \(\sigma_m=\sigma_m(f, D)=\inf \| f -h\| \), where the infimum is taken over all linear combinations \(h=\sum c_lg_l\) with at most \(m\) nonzero coefficients \(c_l\). The authors study the approximation spaces \(A_q^\alpha(D, X)\). By definition, \(A_q^\alpha(D, X)\) consists of all \(f\in X\) for which the norm \[ \| f\| _{A_q^\alpha(D, X)}=\| f\| _X+ \left (\sum_{n=1}^\infty | n^\alpha \sigma_{n-1}| ^q n^{-1}\right )^{1/q} \] is finite. In their earlier paper the authors found relations between the parameters \(\alpha, q, \sigma,\tau\) under which the embedding \(K_q^\tau(D,X) \hookrightarrow A_q^\alpha(D,X)\) takes place, where \(K_q^\tau(D,X)\) is the sparsity spase introduced by DeVore and Temlyakov. Here the authors do the same for the inverse embedding \(A_q^\alpha(D,X) \hookrightarrow K_q^\tau(D,X)\). The main results are for the so-called blockwise incoherent bases in the Hilbert spaces.