We present a direct proof of a known result that the Hardy operator Hf(x) = 1/x ∫0 x f(t) dt in the space L2 = L2(0, ∞) can be written as H = I - U, where U is a shift operator (Uen = en+1, n ∈ Z) for some orthonormal basis {en}. The basis {en} is constructed by using classical Laguerre polynomials. We also explain connections with the Euler differential equation of the first order y' - 1/x y = g and point out some generalizations to the case with weighted Lw 2(a, b) spaces.
@article{urn:eudml:doc:41910,
title = {Structure of the Hardy operator related to Laguerre polynomials and the Euler differential equation.},
journal = {Revista Matem\'atica de la Universidad Complutense de Madrid},
volume = {19},
year = {2006},
pages = {467-476},
zbl = {1121.47025},
mrnumber = {MR2241439},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41910}
}
Kruglyak, Natan; Maligranda, Lech; Persson, Lars-Erik. Structure of the Hardy operator related to Laguerre polynomials and the Euler differential equation.. Revista Matemática de la Universidad Complutense de Madrid, Tome 19 (2006) pp. 467-476. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41910/