The paper concerns the Tikhonov regularized approximation formulae derived by the authors (An and Wu, Inverse Problems 37(1), 015008 (2020)). First, we give more specific an L2 error bound and a uniform error bound of the Tikhonov regularized approximation polynomial. From these two error bounds, we know that a regularization parameter in this approximation polynomial is important. Then, we employ two strategies, the balancing principle and Brezinski– Rodriguez–Seatzu estimators, to select the regularization parameter. Some numerical experiments are given to illustrate that both parameter choice strategies can select suitable regularization parameters. The two parameter choice strategies are also compared by testing some oscillatory functions. Finally, we study the numerical stability of Tikhonov regularized barycentric interpolation formula and Tikhonov regularized modified Lagrange interpolation formula. Both interpolation formulae are shown to be forward stable, and the Tikhonov regularized modified Lagrange interpolation formula is also backward stable. We give numerical examples to illustrate the theoretical results.