On the existence of approximate solution of Fredholm integral equation of the first kind by band-limited scaling function (Q2020160)

The authors consider a Fredholm integral equation of the first kind with convolution type kernel of the form \[ \int_{-\infty }^\infty k(x-y)u(y)\,\mathrm{d}y=g(x)\,,\quad x\in \mathbb{R}\,, \] where the kernel \(k\) and the data \(g\) are known functions whereas \(u\) has to be determined. The solution of this integral equation is approximated via band-limited scaling functions generated by a class of band-limited wavelets. Existence and uniqueness of the approximated solution is established and shown to be rapidly converging with a significant reduction in the computational cost. Some numerical results are provided to support their theoretical findings.