Classical Gaussian formulas are well known. They construct a polynomial interpolating in the zeros of a polynomial orthogonal with respect to a positive measure \(\alpha\) and the integral of this polynomial is a quadrature formula for \(\int f(x) d\alpha(x)\) with maximal polynomial degree of exactness. The generalized Gaussian formulas considered in this paper allow the interpolation points to coincide and a Hermite interpolating polynomial is used instead. The knots are generalized Chebyshev points in the sense that the the monic polynomial with these zeros (multiplicity imposed) has minimal \(L_1(d\alpha)\) norm in \([-1,1]\). Under appropriate conditions the convergence of these generalized Gaussian quadrature formulas is proved and error estimates are derived.