Gauss-type formulas for linear functionals

We give a method, by solving a nonlinear system of equations, for Gauss harmonic interpolation formulas which are useful for approximating, the solution of the Dirichlet problem.

We also discuss approximations for integrals of the form

I(f) = (1/2πi) ∫_L (f(z)/(z-α)) dz.

Our approximations shall be of the form

Q(f) = Σ_k=1ⁿ A_kf(τ_k).

We characterize the nodes τ₁, τ₂, …, τ_n, to get the maximum precision for our formulas.

Finally, we propose a general problem of approximating for linear functionals; our results need further development.