Embeddings for general weighted sequence spaces and entropy numbers (Q2707678)

This paper deals with the estimates of entropy number for imbeddings of Besov spaces with generalized smoothness in the critical and subcritical case. The author uses the Fourier analysis approach to the Besov spaces, hence they appear in the form of sequence spaces \(\ell _q(\beta _j^{M_j})\), consisting of all \((x_{j,k})_{j\in \mathbb N, k=1,\dots ,M_j}\) such that \((\sum _{j=0}^{\infty }\beta _j^q (\sum _{k=1}^{M_j}|x_{j,k}|^{p})^{q/p})^{1/q}<\infty \). Here \(M_j\) are positive integers, and \(\beta _j\) is a sequence of positive real numbers satisfying the growth condition \(d_0\beta _j\leq \beta _{j+1}\leq d_1\beta _j\) for fixed constants \(d_0\) and \(d_1\) and for every \(j\). The resulting spaces generalize usual Besov spaces and several examples, as spaces with logarithmic smoothness, are given. The author proves estimates for the entropy numbers for the cases \(0<p_1\leq p_2\) and \(p_1=p_2\) (the first parameters of Besov spaces related to the integration properties) in the subcritical case, and for the critical case he considers \(0<p_1<p_2\), and \(\beta _j=(1+j)^b\).NEWLINENEWLINEFor the entire collection see [Zbl 0933.00035].