Piecewise monotone spline interpolation

Abstract Let ( x i , y i ), i = 0, 1,…, k , be a set of points, with x 0 x 1 x k . We prove the existence of a spline function of specified deficiency, f ( x ), which satisfies f ( x i ) = y i , i = 0, 1,…, k , and which is monotone on each of the intervals [ x i − 1 , x i ], i = 1, 2,…, k .