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.