Application of Chebyshev polynomials to the regularization of ill-posed and ill-conditioned equations in Hilbert space

Abstract

WE consider in Hilbert space the equation Ax = f, where 0< A⩽ E, and only the approximation f δ of f, ∥f δ− f∥⩽ δ. is known. We select a polynomial P n ( λ), which is expressed simply in terms of the Chebyshev polynomials T n + in1 and approximates 1 λ , on [0, 1] fairly well, in the sense that the values of P n ( λ) are not too great on [0, ε] and are close to 1 λ , on [ε, 1], where ε is a small parameter. The approximate solution is represented in the form x δen = P n ( A) fδ. An estimate of the error is given.