Lattice points can be extended over particular regions in R2 (see [ 25] for the case of constant weight) . More see [ 9] ) , let A be an arbitrary lattice in R2. Let g R2 be a convex region containing the origin and possessing a boundary curve aG such that its normal field v is continuously differentiable and its curvature is nonvanishing. Suppose that F is of class C( 2) ( C) . Then, for all a, y E R2, the series