Simultaneous approximation of Hilbert and Hadamard transforms on bounded intervals (Q6572623)

This interesting paper under review studies the simultaneous approximation of the weighted Hilbert and Hadamard transforms of a fixed function \(f:(-1,1)\to \mathbb R\) with suitable smoothness and typically satisfying \(|(fw)(x)|\to 0,\, x\to \pm 1\) where the weight \(w\) is a Jacobi weight with certain parameters. Given such a fixed function \(f\) and weight \(w\), define the weighted Hilbert transform of \(f\) and weighted Hadamard transform of \(f\) respectively by\N\[\NH_0^w(f,x):=\lim_{\varepsilon\to 0}\int_{|t-x|>\varepsilon}\frac{f(t)w(t)}{t-x}dt,\, x\in (-1,1).\N\]\N\[\NH_1^w(x):=\frac{d}{dx}H_0^w(f,x).\N\]\NThese operators have numerous applications in areas diverse as integral equations, signal processing, data science, harmonic analysis and approximation theory. One classical approximation technique is to approximate well the function \(f\) by a Lagrange interpolation polynomial of certain degree where the interpolation is with respect to the zeroes of the orthonormal polynomials with respect to the weight \(w\) and then applying a scheme of product integration rules. The authors propose a scheme of product integration rules for the simultaneous approximation of \(H_0^w\) and \(H_1^w\) for different classes of functions \(f\) and Jacobi weights \(w\). The advantages of such a scheme are for example a saving in the number of function evaluations and the avoidance of the derivatives of the density function \(f\) when approximating the Hadamard transform. Stability and convergence results are obtained and theoretical estimates are confirmed by numerical tests.\N\NThe paper is well written.