BIT 22 (1982), 331-338 CONVEX INTERPOLATING SPLINES OF ARBITRARY DEGREE II EDWARD NEUMAN Institute of Computer Science, University of Wroclaw, uL Przesmyckie#o 20, 51-151 Wroclaw, Poland Abstract. For given data {(x~, Yl)}~'= o, (Xo < xl <-.. < x,) we consider the possibility of finding a spline function s of arbitrary degree k + 1 (k > 1) with preassigned smoothness l, where 1 < l < [(k + 1)/2]. The spline s should be such that s(x~) = y~, i = 0, 1..... n and s is increasing and convex on [Xo, x~]. Sufficient conditions which guarantee the existence of s and an explicit formula for this function are derived. 1. Introduction and notation. Let l-a, b] be a given compact interval on the real line R, and let An denote an arbitrary but fixed partition of this interval with knots xi, i.e. An:a = x o < x I < ... < x n = b. Further let Sp(k + 1, l, An) (k,l ~ No;0 < l < k) be the set of polynomial splines of degree k + 1 and deficiency equal to k + 1 - I. That is, s ~ Sp(k + 1, l, An) ifs ~ Ct[a, b] and on each of the intervals [x~_ 1, xJ (i = 1, 2.... , n) s coincides with an algebraic polynomial of degree k + 1 or less. It is a well-known fact that Sp(k + 1, l, An) is a linear subspace of C[a, b], and dim Sp(k + 1, l, An) = n(k + 1 - l) + l + 1. For fixed k and I (0 < l < k; k > 1) we ask: Does there exist s6Sp(k+l,I,A,) satisfying the following conditions (1.1) s(xi) = Yl (i = 0, 1 ..... n; Yi given reals), and (1.2) s is convex and increasing on [a, b] ? In this paper we restrict our attention to the case when 1 < l < [(k + 1)/2], where as usual I-q] denotes the greatest integer < q. Also we assume that k > 1 (for k = 0 our problem is a trivial one). This paper is a continuation of the paper [7], where this problem was solved in the case I = k. For related results see also [3-6] and [9-10]. Received Dec. 9, 1981.