×

Entropy numbers of certain summation operators. (English) Zbl 1042.47011

Let \(a=(\alpha_k)_{k=-\infty}^\infty\) and \(b=(\beta_k)_{k=-\infty}^\infty\) be two sequences of non-negative real numbers. Then one defines (whenever it exists) the weighted summation operator \(S_{a,b}\) by \[ S_{a,b}(x):= \left(\alpha_k\sum_{l<k}\beta_l x_l\right)_{k=-\infty}^\infty\;,\qquad x=(x_k)_{k=-\infty}^\infty\;. \] The aim of the present paper is to characterize sequences \(a\) and \(b\) for which \(S_{a,b}\) is a bounded operator from \(l_p(\mathbb Z)\) into \(l_q(\mathbb Z)\) with \(1\leq p,q\leq\infty\). Furthermore, if \(S_{a,b}\) is even compact, then optimal estimates for the entropy numbers of \(S_{a,b}\) (in terms of quantities defined by \(a\) and \(b\)) are proved. The results are applied to small deviation problems for weighted random sequences \((W(t_k))_{k=-\infty}^\infty\) for some increasing \(t_k>0\) and the Wiener process \(W\).

MSC:

47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
60G15 Gaussian processes
PDFBibTeX XMLCite
Full Text: EuDML