Structure of the Hardy operator related to Laguerre polynomials and the Euler differential equation.

Kruglyak, Natan, Maligranda, Lech, and Persson, Lars-Erik. "Structure of the Hardy operator related to Laguerre polynomials and the Euler differential equation.." Revista Matemática Complutense 19.2 (2006): 467-476. <http://eudml.org/doc/41910>.

@article{Kruglyak2006,
abstract = {We present a direct proof of a known result that the Hardy operator Hf(x) = 1/x ∫0x 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 Lw2(a, b) spaces.},
author = {Kruglyak, Natan, Maligranda, Lech, Persson, Lars-Erik},
journal = {Revista Matemática Complutense},
keywords = {Desigualdad de Hardy; Espacios de Lebesgue; Isometría; Polinomios de Laguerre; Ecuación de Euler; Teoría de operadores; Hardy inequalities; Hardy operator; Laguerre polynomials; basis},
language = {eng},
number = {2},
pages = {467-476},
title = {Structure of the Hardy operator related to Laguerre polynomials and the Euler differential equation.},
url = {http://eudml.org/doc/41910},
volume = {19},
year = {2006},
}

TY - JOUR
AU - Kruglyak, Natan
AU - Maligranda, Lech
AU - Persson, Lars-Erik
TI - Structure of the Hardy operator related to Laguerre polynomials and the Euler differential equation.
JO - Revista Matemática Complutense
PY - 2006
VL - 19
IS - 2
SP - 467
EP - 476
AB - We present a direct proof of a known result that the Hardy operator Hf(x) = 1/x ∫0x 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 Lw2(a, b) spaces.
LA - eng
KW - Desigualdad de Hardy; Espacios de Lebesgue; Isometría; Polinomios de Laguerre; Ecuación de Euler; Teoría de operadores; Hardy inequalities; Hardy operator; Laguerre polynomials; basis
UR - http://eudml.org/doc/41910
ER -