Positive results and counterexamples in comonotone approximation II

D. Leviatan*, I. A. Shevchuk, O. V. Vlasiuk

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Let E n (f) denote the degree of approximation of f ∈ C [ - 1, 1] , by algebraic polynomials of degree <n, and assume that we know that for some α > 0 and N ≥ 2, nαEn(f)≤1,n≥N. Suppose that f changes its monotonicity s ≥ 1 times in [ - 1, 1] We are interested in what may be said about its degree of approximation by polynomials of degree <n that are comonotone with f. In particular, if f changes its monotonicity at Y s {colon equals} {y1, . . . , y s} and the degree of comonotone approximation is denoted by E n (f, Y s) , we investigate when can one say that nαEn(f,Ys)≤c(α,s,N),n≥N*, for some N *. Clearly, N *, if it exists at all (we prove it always does), depends on α, s and N. However, it turns out that for certain values of α, s and N, N * depends also on Y s and in some cases even on f itself. The results extend previous results in the case N = 1.

Original languageEnglish
Pages (from-to)1-23
Number of pages23
JournalJournal of Approximation Theory
StatePublished - Feb 2014


  • Comonotone polynomial approximation
  • Degree of approximation
  • Uniform norm


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