Nyström interpolants based on zeros of Laguerre polynomials for some Weiner-Hopf equations

We consider a special class of Wiener-Hopf integral equations $$u(y)-\lambda\int_0^\infty{h(y-x)}u(x)\text{?}dx=\text{?}f(y),$$ with ${||\int_0^\infty{\vert{h(y-x)}\vert}dx||}_{\infty}<\lambda^{-1}$ , whose solutions u(x) decay exponentially at infinity. First we rewrite the equation in the form $$v(y)-\mu \mathrm{e}^y\int_0^\infty{\mathrm{e}^{-x}k(y-x)}v(x)\text{?}dx=g(y),$$ with ${|| \mathrm{e}^y\int_0^\infty \mathrm{e}^{-x}{\vert{k(y-x)\mathrm{d}x}\vert}|}_\infty < \mu^{-1}$ . Then we propose to solve this equation by using a Nyström interpolant based on a product rule of the type $$\int_0^\infty \mathrm{e}^{-x}k(y-x)v(x)\mathrm{d}x\;\sum_{t=1}^m{\omega_{mi}(y)v(x_{mi})},$$ where the nodes {x^m1} are the zeros of the m-degree Laguerre polynomial. For these interpolants we prove stability in C[0, ∞) and derive uniform convergence estimates.