Solving dual integral equations on Lebesgue spaces

Abstract

We study dual integral equations associated with Hankel transforms, that is, dual integral equations of Titchmarsh's type. We reformulate these equations giving a better description in terms of continuous operators on L p spaces, and we solve them in these spaces. The solution is given both as an operator described in terms of integrals and as a series ∞ n=0 c n J µ+2n+1 which converges in the L p-norm and almost everywhere, where J ν denotes the Bessel function of order ν. Finally, we study the uniqueness of the solution.

The multiplier of the interval [−1,1] for the Dunkl transform on the real line

“…Regarding the weighted boundedness of this multiplier, it has shown to be useful to deal with some practical problems, as in the resolution of dual integral equations of Titchmarsh's type; for details, see [1].…”

Discrete Fourier–Neumann series

“…Let U 1 and U 2 be the bilinear operators on l p ðNÞ Â l q ðNÞ given by As a standard consequence of Theorem 1, we obtain that S n b-b; in l p ðNÞ; for all bAb p;a ; if 4 3 opo4 and aX À 1 2 : This will be more interesting if we can describe the spaces b p;a : We do it in this section.…”

“…Let aX À 1 2 and 1opoN: Then there exists a constant C independent of n and b such that jjS n bjj l p ðNÞ pCjjbjj l p ðNÞ ; 8bAl p ðNÞ 3 4 3 opo4:…”

“…& Proof (Proof of Theorem 1). First, we are going to prove that 4 3 opo4 is a necessary condition for the uniform boundedness of S n :…”

“…Fourier-Neumann series have had a prominent role in the study of band-limited functions for the Fourier transform (see [1]) and in the analysis of dual integral equations (see [4]). Moreover, some of the operators appearing in Fourier-Neumann expansions are related to the disc multiplier for the Fourier transform (see [2,5]).…”