Two-sided estimates of the approximation numbers of certain Volterra integral operators

Edmunds, D., Evans, W., and Harris, D.. "Two-sided estimates of the approximation numbers of certain Volterra integral operators." Studia Mathematica 124.1 (1997): 59-80. <http://eudml.org/doc/216397>.

@article{Edmunds1997,
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 (an(T)) when 1 < α < ∞.},
author = {Edmunds, D., Evans, W., Harris, D.},
journal = {Studia Mathematica},
keywords = {Volterra integral operator; estimates for the approximation number},
language = {eng},
number = {1},
pages = {59-80},
title = {Two-sided estimates of the approximation numbers of certain Volterra integral operators},
url = {http://eudml.org/doc/216397},
volume = {124},
year = {1997},
}

TY - JOUR
AU - Edmunds, D.
AU - Evans, W.
AU - Harris, D.
TI - Two-sided estimates of the approximation numbers of certain Volterra integral operators
JO - Studia Mathematica
PY - 1997
VL - 124
IS - 1
SP - 59
EP - 80
AB - 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 (an(T)) when 1 < α < ∞.
LA - eng
KW - Volterra integral operator; estimates for the approximation number
UR - http://eudml.org/doc/216397
ER -