MATHEMATICS OF COMPUTATION, VOLUME 30, NUMBER 136 OCTOBER 1976, PAGES 765-771 A Bound on the L, -Norm of L -Approximation by Splines in Terms of a Global Mesh Ratio By Carl de Boor* Abstract. Let Lkf denote the least-squares approximation to /S Lj by splines of n+k order k with knot sequence t = (t¡)x ■ In connection with their work on Galerkin's method for solving differential equations, Douglas, Dupont and Wahlbin have shown that the norm Hi-^ll,,, of Lk as a map on L«, can be bounded as follows, Halloo < constkMt, with Aij a global mesh ratio, given by Mx := max A^./min {Af,.|Ai; > o}. Using their very nice idea together with some facts about S-splines, it is shown here that even IILfclU « constk(M\k))'/2 with the smaller global mesh ratio AÍJ ' given by M<fc) := max (t¡+k - (,)/<*/+* - tj). Ui A mesh independent bound for L2-approximation by continuous piecewise poly- nomials is also given. 1. Introduction. This note is an addendum to the clever paper by Douglas, Du- pont and Wahlbin [2] in which these authors bound the Unear map of least-squares approximation by splines of order k with knot sequence t := (/,), as a map on L^, in terms of the particular global mesh ratio Mt := max Ar/min {At¡\At¡> 0}. Their argument is very elegant. But their result is puzzling in one aspect: The ratio Mt is not a continuous function of t. If, e.g., t is uniform, hence Aft = 1, and we now let t —► t* by letting just one knot approach its neighbor, leaving all other knots fixed, then Urn Mt = °°, while Mt« = 2. t-t* l Correspondingly, their bound goes to infinity as t —► t*, yet is again finite for the particular knot sequence t*. This puzzling aspect is removed below. It is shown that (as asserted in a footnote to [1]) their very nice argument can be used to give a bound in terms of the smaUer global mesh ratio (1) M<k> := max (ti+k - r,.)/min (ti+k - t¡) Received November 10, 1975 AMS (MOS) subject classifications (1970). Primary 41A15. Key words and phrases. Least-squares approximation by splines, error bounds. 'Sponsored by the United States Army under Contract No. DAAG29-75-C-0024. Copyright © 1976. American Mathematical Society 765 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use