Characterization of generalized convex functions by best L2-approximations

Dan Amir*, Zvi Ziegler

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Let M be a normed linear space, and {Mn}1 a sequence of increasing finite dimensional subspaces, i.e., Mn ⊂ Mn + 1, for all n. For any element f{hook} ε{lunate} M, we obviously have d(f{hook}, Mn) ≥ d(f{hook}, Mn + 1), for all n, (1) where d(f{hook}, Mk) is the distance, in the metric induced by the norm, from Mk to f{hook}. In a recent paper [2], we discussed the space M = C[a,b] with the uniform norm and with Mn = [u0,..., un - 1], the linear subspace spanned by {ui}0n - 1, where {ui}0 is an infinite Tchebycheff system. We established there that the functions for which inequality (1), for a given n, is strict for all subintervals of [a, b] are precisely those that are convex with respect to (u0, u1,..., un - 1). The proof depended crucially on the alternance properties of the best approximants in the uniform norm. Somewhat surprisingly, analogous results are valid when the norm under consideration is the L2-norm. In fact, as we show in this paper, generalized convex functions play the same role in the L2-norm, for all continuous weights. Weaker results for the L2 case were obtained in [6].

Original languageEnglish
Pages (from-to)115-127
Number of pages13
JournalJournal of Approximation Theory
Issue number2
StatePublished - Jun 1975


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