On solving some Cauchy singular integral equations by de la Vallée Poussin filtered approximation (Q6546958)

Using orthonormal Jacobi polynomials and corresponding de la Vallée Poussin polynomial quasi-projections for the Jacobi weight \(u(x) = (1 - x)^{\alpha}(1 + x)^{-\alpha}\) with \(x \in (-1,\,1)\) and \(0 < |\alpha| < 1\), the authors discuss the numerical solution of the Cauchy singular integral equation \N\[\NDf(y) + \nu\,Kf(y) = g(y)\,, \quad y \in (-1,\,1)\,, \N\]\Nwhere \(\nu \in \mathbb R\) and \(g\) are given and the operators are defined by \N\[\NDf(y) = \cos(\pi \alpha)\,f(y)\,u(y) - \frac{\sin(\pi \alpha)}{\pi}\,\int_{-1}^1 \frac{f(x)}{x-y}\,u(x)\,\mathrm{d}x\,,\quad Kf(y) = \int_{-1}^1 k(x,y)\,f(x)\,u(x)\,\mathrm{d}x\,.\N\]\NThe numerical method proposed is convergent and stable, and provides a near-best polynomial approximation of the solution \(f\) by solving a well-conditioned linear system. Numerical tests show a better local approximation of \(f\) compared with the classical Lagrange interpolation method.