Pointwise estimates for monotone polynomial approximation

Abstract

Abstract. We prove that if f is increasing on [ -1,1], then for each n = 1, 2 ..... there is an increasing algebraic polynomial P. of degree n such that {f(x) -P.(x){ < cw2( f, V/I -x 2/n), where w2 is the second-order modulus of smoothness. These results complement the classical pointwise estimates of the same type for unconstrained polynomial approximation. Using these results, we characterize the monotone functions in the generalized Lipschitz spaces through their approximation properties.

Pointwise Estimates for Convex Polynomial Approximation

“…Recently Shvedov [10] has extended these results by showing that for a monotone / G C[-í, 1] there are monotone polynomials pn of degree < n such that (2) ||/-Pn||<Cw2(/,l/n) where u2(f,-) is the second modulus of smoothness of /. Moreover, he has proved that one cannot expect (2) to hold with W3 replacing u/2.…”

“…Moreover, he has proved that one cannot expect (2) to hold with W3 replacing u/2. Shvedov [9,10] also discussed the question of convex polynomial approximation to convex functions / G C [-1,1] showing that there exist convex polynomials pn satisfying (2).…”

“…Recently DeVore and Yu [2] have constructed a sequence of monotone polynomials pn associated with a monotone function / G C[-1,1] and yielding the following pointwise approximation rate:…”

“…Then the author [6] has shown that the DeVore and Yu polynomials [2] also yield uniform monotone approximation at the rate…”

“…In this note we will modify the DeVore-Yu polynomials and for a convex /, obtain polynomials which are convex on [-1,1] and which satisfy (3) and (4), thus improving Shvedov's estimates (2). Also, if in addition, / is monotone, then the polynomials are monotone.…”