Collocation with Chebyshev polynomials for a hypersingular integral equation on an interval (Q1195732)

The hypersingular integral operator \(D\) defined by \(Dv(x)=- (1/\pi)\text{p.f.} \int_{-1}^ 1 v(y)(x-y)^{-2}dy\), \(x\in[-1,1]\), has the property \(D(\omega U_ j)=(j+1)U_ j\), \(j=0,1,2,\dots,\) where \(\omega(x)=(1-x^ 2)^{1/2}\) and \(U_ j\) is the Chebyshev polynomial of the second kind. The authors use this property to design and analyse the convergence rate of a collocation method for the equation \(Dv=g\) with a given smooth function \(g\).