Convex interpolating splines of arbitrary degree II

For given data {(x _i ,y _i )} _i=0 ⁿ , (x ₀<x ₁<...<x _n ) we consider the possibility of finding a spline functions of arbitrary degreek+1 (k 1) with preassigned smoothnessl, where 1 l (k+1)/2]. The splines should be such thats(x _i )=y _i ,i=0, 1,...,n ands is increasing and convex on x ₀,x _n ]. Sufficient conditions which guarantee the existence ofs and an explicit formula for this function are derived.