Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms

Abstract Chebyshev polynomials of the third and fourth kinds, orthogonal with respect to (1 + x ) 1 2 (1 − x ) − 1 2 and (1 − x ) 1 2 (1 + x ) − 1 2 , respectively, on [− 1, 1], are less well known than traditional first- and second-kind polynomials. We therefore summarise basic properties of all four polynomials, and then show how some well-known properties of first-kind polynomials extend to cover second-, third- and fourth-kind polynomials. Specifically, we summarise a recent set of first-, second-, third- and fourth-kind results for near-minimax constrained approximation by series and interpolation criteria, then we give new uniform convergence results for the indefinite integration of functions weighted by (1 + x ) − 1 2 or (1 − x ) − 1 2 using third- or fourth-kind polynomial expansions, and finally we establish a set of logarithmically singular integral transforms for which weighted first-, second-, third- and fourth-kind polynomials are eigenfunctions.