Pointwise and uniform approximation of the Hilbert transform

The Hilbert transform of a function g, H(g) is an important tool in many mathematical fields. Expecially its numerical evaluation is often useful in some procedures for searcing solutions of the singular integral equations. In this context an approximation of (HV^alpha,beta,f;t), |t|1, where f is a continuous function in [-1,1] and v^alpha,beta, alpha,beta>-1 is a Jacobi weight, is required. In the last decade more then one paper appeared on this subject and among others we recall [1,2,3,4,5,14,15,20]. The procedure used in these papers can be described as follows. Approximate the function f with a Lagrange interpolating polynomial and obtain a quadrature formula with coefficients depending on the singularity t. The error of the quadrature formula was estimated assuming t in a "more or less" wide neighborhood of zero since (HV^alpha,beta,f;t) is unbounded at the endpoints of (-1,1). In this paper we propose a more accurate procedure: since (HV^alpha,beta,f;t) is unbounded at the endpoints, itis more natural to consider its approximation in some functional spaces equipped with weighted uniform norms. In Section 2 (see theorems 2.1 and 2.2) we state the behaviour of (HV^alpha,beta,f;t) in [-1,1] and we give new conditions for its exstence. In sections 3 and 4 we cosntruct approximations of (HV^alpha,beta,f;t) which converge to the exact value in weighted uniform norm. The L^p convergence of the formula is also briefly treated. Finally since the considered procedures are numerically stable, these can be used for the numerical evaluation of (HV^alpha,beta,f;t) . In this sense the present paper is a short survey on the subject.