MATHEMATICS OF COMPUTATION Volume 72, Number 244, Pages 1873–1885 S 0025-5718(03)01523-0 Article electronically published on May 1, 2003 ON THE PROBLEMS OF SMOOTHING AND NEAR-INTERPOLATION SCOTT N. KERSEY Abstract. In the first part of this paper we apply a saddle point theorem from convex analysis to show that various constrained minimization problems are equivalent to the problem of smoothing by spline functions. In particular, we show that near-interpolants are smoothing splines with weights that arise as Lagrange multipliers corresponding to the constraints in the problem of near-interpolation. In the second part of this paper we apply certain fixed point iterations to compute these weights. A similar iteration is applied to the computation of the smoothing parameter in the problem of smoothing. 1. Introduction Let X := L (m) 2 ([a, b]−→R) denote the Sobolev space of functions f :[a, b]−→R whose derivatives D j-1 f are in the Lebesgue spaces L 2 ([a, b]−→R) for j =1:m, m ≥ 1. Let t be a sequence of data sites a = t 1 ≤ t 2 ≤ ··· ≤ t n = b with multiplicity at most m, i.e., m i := #{t k : k ≤ i and t k = t i }≤ m for i=1:n. For these fixed data sites, let Λ be the data map Λ: f −→ Λ f =(λ i f ) with λ i : f −→ D mi-1 f (t i ). In particular, λ i f = f (t i ) when m i = 1. Given a sequence of data z ∈ R n and nonnegative weights w ∈ R n ≥0 , we define the functionals J : f −→  b a |D m f (u)| 2 du, E i : f −→ |λ i f − z i |, E w : f −→  n  i=1 w i E i (f ) 2  1/2 . Let ρ> 0 (the smoothing parameter ), ε ∈ R n >0 (a sequence of tolerances ε i ), S> 0, and let q and r in R n be such that q i <r i for all i. We are interested in the following problems: (A) minimize f ∈X J (f )+ ρE w (f ) 2 , (B) minimize f ∈X {J (f ): E i (f ) ≤ ε i ,i=1:n}, (C) minimize f ∈X {J (f ): E w (f ) ≤ S}, (D) minimize f ∈X {J (f ): q i ≤ λ i f ≤ r i ,i=1:n}. Received by the editor July 20, 1999 and, in revised form, September 21, 2001. 2000 Mathematics Subject Classification. Primary 41A05, 41A15, 41A29. Key words and phrases. Near-interpolation, smoothing splines, approximation. c 2003 American Mathematical Society 1873 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use