On polynomial interpolation of intervals

Interpolation methods have been very popular in many practical problems. To estimate a function value y for a given x, we need to measure several pairs of (x_i, y_i). In some problems, we do not know the exact value of the measured quantities, due to the inevitable measurement inaccuracy. While, we know the ranges of the measured quantities; namely, we represent the quantities as intervals. In this paper, we study the polynomial interpolation of intervals. First, a general form of the interval coefficients polynomial is given. Then, the existence theorem of interval interpolating polynomial is proved. We give two solution formulas, in Lagrange form and Newton form, and get the relations of them. Finally, two examples are provided to illustrate the rationality of the method and the validity of the solution.