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

Natan Kruglyak; Lech Maligranda; Lars-Erik Persson

Revista Matemática Complutense (2006)

- Volume: 19, Issue: 2, page 467-476
- ISSN: 1139-1138

## Access Full Article

top## Abstract

top## How to cite

topKruglyak, 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 -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.