The Erd Under A Generalized Freud-type Weights (?) S Interpolation Convergence Process

We investigate a convergence process in the weighted polynomial interpolation and results apply to the zeros of orthonormal polynomials {p_n(w_rQ²,x)}_n=0^∞ as the nodes of interpolation X = {x_kn}_k=1ⁿ associated with a generalized Freud-type-weight w_rQ²(x), that is, w_rQ²(x) = |x|^2r exp(-2Q(x)), where r ≥ 0 and Q : R â†' R is even and continuous, (R = (+∞,-∞)), Q'(x) is continuous, Q'(x) > 0, (?)x ∈ (0,+∞), and Q" is continuous in (0,+∞) and furthermore, Q satisfies further conditions. To begin with, we derive the behaviour of the corresponding weighted Lagrange interpolatory Lebesgue number ∧_n(w,X), whose weak approximation order is n^1/6 by abuse of the properties of {p_n(w², x)}_n=0^∞. In this case, considering that Lebesgue number is no less than log n, we modify the Lagrange interpolatory point group suitably and turn out that the weak approximation order of ∧_n(w, X) may attain log n for the modified Lagrange weighted interpolation. Moreover, on the basis of this, we obtain the weighted Erdos-type convergence theorem in the interpolation at such point system.