  This paper is concerned with the introduction of Tikhonov regularization into
least squares approximation scheme on $[-1,1]$ by orthonormal polynomials, in
order to handle noisy data. This scheme includes interpolation and
hyperinterpolation as special cases. With Gauss quadrature points employed as
nodes, coefficients of the approximation polynomial with respect to given basis
are derived in an entry-wise closed form. Under interpolatory conditions, the
solution to the regularized approximation problem is rewritten in forms of two
kinds of barycentric interpolation formulae, by introducing only a
multiplicative correction factor into both classical barycentric formulae. An
$L_2$ error bound and a uniform error bound are derived, providing similar
information that Tikhonov regularization is able to reduce the operator norm
(Lebesgue constant) and the error term related to the level of noise, both by
multiplying a correction factor which is less than one. Numerical examples show
the benefits of Tikhonov regularization when data is noisy or data size is
relatively small.
