| The present work joints some results about Fourier-Neumann series and its application in solving dual integral equations. These series are defined using the orthonormal system in L20,infinity ,xadx given, for a > -1, by janx =a+2n+1Ja+2n+1 xx -a+1/2, n=0,1,2&ldots;, where Jmu denotes the Bessel function of order mu. So, for an appropriate function f, we define its Fourier-Neumann series as f∼k=0infinitya kfjk, withakf =0infinityf tjak ttadt.; After some previous technical results about Bessel functions, we analyze some uniform boundedness for the partial sum operator, defined by Snf=k=0na kfja k . In a more precise form, we study a weighted inequality (or boundedness from Lp0,infinity ,upxxa dx into itself, with u(x) = xa(1 + x)b-a ), the weak boundedness (or boundedness from Lp0,infinity ,xadx into Lp,infinity0,infinity ,xadx and the restricted weak boundedness (i.e., the weak boundedness for f = chiE, with E<infinity ).; From the uniform boundedness for Sn in Lp0,infinity ,xadx , we deduce that Snf → f, in Lp0,infinity ,xadx , for each function f∈Bp,a , where Bp,a=spanj an , with clausure in Lp0,infinity ,xadx . We describe a generalization of the spaces Bp,a in one of our results. Taking the Hankel transform of order a as the operator Haf,x =x-a/22 0infinityft Jaxt ta/2dt,x>0, the spaces Bp,a and its generalization are characterized using the chi[0,1] -multiplier for the Hankel transform.; We use Fourier-Neumann series for solving dual integral equations in Lebesgue spaces. This problem can be stated in the following way: given a function g, defined on [0,1], we look for f, defined on (0, infinity ), such that 0infinityt bftJa xtdt=g x,if 0<x<1, 0infinityft Jaxt dt=0,if x>1. We show that the function f can be expressed as a Fourier-Neumann series in which the coeficients depend on g.; Finally, a uniform boundedness for the Bochner-Riesz multiplier of the Hankel transform is obtained. |
