Published Volumes RMC
Volume 19 num. 2
Structure of the Hardy Operator Related to
Laguerre Polynomials and the Euler
Dierential Equation
Natan KRUGLYAK, Lech MALIGRANDA,
and Lars-Erik PERSSON
and Lars-Erik PERSSON
Department of Mathematics
Luleå University of Technology
SE-971 87 Luleå — Sweden
Luleå University of Technology
SE-971 87 Luleå — Sweden
Accepted: March 21, 2006
ABSTRACT
We present a direct proof of a known result that the Hardy
operator
in the space
can be written as
, where
is a shift operator
,
for some
orthonormal basis
. The basis 
is constructed by using classical
Laguerre polynomials. We also explain connections with the Euler
differential
equation of the first order
and point out some generalizations to
the case with weighted
spaces.
Key words: Hardy inequality, Hardy operator, Laguerre polynomials, isometry,
Lebesgue spaces, basis in
space, weighted
spaces.
2000 Mathematics Subject Classication: 47B38.