Weighted mean convergence of Hakopian interpolation on the disk

In this paper, we study weighted mean integral convergence of Hakopian interpolation on the unit disk D. We show that the inner product between Hakopian interpolation polynomial H_n(f;x,y) and a smooth function g(x,y) on D converges to that of f (x,y) and g(x,y) on D when n → 8, provided f (x,y) belongs to C(D) and all first partial derivatives of g(x,y) belong to the space Lip _M ^α (0 < α ≤ 1). We further show that provided all second partial derivatives of g(x,y) also belong to the space Lip _M ^α and f(x,y) belongs to C ¹(D), the inner product between the partial derivative of Hakopian interpolation polynomial ∂/∂x H_n (f;x,y) and g(x,y) on D converges to that between ∂/∂x f(x,y) and g(x,y) on D when n→∞.