Two-sided estimates of the approximation numbers of certain Volterra integral operators
Volume 124 / 1997
Studia Mathematica 124 (1997), 59-80
DOI: 10.4064/sm-124-1-59-80
Abstract
We consider the Volterra integral operator $T:L^{p}(ℝ^{+}) → L^{p}(ℝ^{+})$ defined by $(Tf)(x) = v(x)ʃ_{0}^{x} u(t)f(t)dt$. Under suitable conditions on u and v, upper and lower estimates for the approximation numbers $a_n(T)$ of T are established when 1 < p < ∞. When p = 2 these yield $lim_{n→∞} na_{n}(T) = π^{-1} ʃ_{0}^{∞} |u(t)v(t)|dt$. We also provide upper and lower estimates for the $ℓ^{α}$ and weak $ℓ^{α}$ norms of (a_{n}(T)) when 1 < α < ∞.
