Uniform convergence for sequences of best L^{p} approximation (original) (raw)

Interpolatory pointwise estimates for monotone polynomial approximation

Journal of Mathematical Analysis and Applications, 2018

Given a nondecreasing function f on [−1, 1], we investigate how well it can be approximated by nondecreasing algebraic polynomials that interpolate it at ±1. We establish pointwise estimates of the approximation error by such polynomials that yield interpolation at the endpoints (i.e., the estimates become zero at ±1). We call such estimates "interpolatory estimates". In 1985, DeVore and Yu were the first to obtain this kind of results for monotone polynomial approximation. Their estimates involved the second modulus of smoothness ω2(f, •) of f evaluated at √ 1 − x 2 /n and were valid for all n ≥ 1. The current paper is devoted to proving that if f ∈ C r [−1, 1], r ≥ 1, then the interpolatory estimates are valid for the second modulus of smoothness of f (r) , however, only for n ≥ N with N = N(f, r), since it is known that such estimates are in general invalid with N independent of f. Given a number α > 0, we write α = r + β where r is a nonnegative integer and 0 < β ≤ 1, and denote by Lip * α the class of all functions f on [−1, 1] such that ω2(f (r) , t) = O(t β). Then, one important corollary of the main theorem in this paper is the following result that has been an open problem for α ≥ 2 since 1985: If α > 0, then a function f is nondecreasing and in Lip * α, if and only if, there exists a constant C such that, for all sufficiently large n, there are nondecreasing polynomials Pn, of degree n, such that |f (x) − Pn(x)| ≤ C √ 1 − x 2 n α , x ∈ [−1, 1].

N ov 2 01 7 Interpolatory pointwise estimates for monotone polynomial approximation

2018

Polynomial interpolation, an L-function, and pointwise approximation of continuous functions

Journal of Approximation Theory, 2008

We show that if {s k } ∞ k=1 is the sequence of all zeros of the L-function L(s,) := ∞ k=0 (−1) k (2k + 1) −s satisfying Re s k ∈ (0, 1), k = 1, 2,. .. , then any function from span {|x| s k } ∞ k=1 satisfies the pointwise rapid convergence property, i.e. there exists a sequence of polynomials Q n (f, x) of degree at most n such that f − Q n C[−1,1] C(f)E n (f), n=1, 2,. .. , and for every x ∈ [−1, 1], lim n→∞ (|f (x)−Q n (f, x)|)/E n (f)= 0, where E n (f) is the error of best polynomial approximation of f in C[−1, 1]. The proof is based on Lagrange polynomial interpolation to |x| s , Re s > 0, at the Chebyshev nodes. We also establish a new representation for |L(x,

Monotone polynomial approximation in LpL^pLp

Rocky Mountain Journal of Mathematics, 1989

Jackson type estimates on the rate of approximation of monotone functions in L p [-1,1] by means of monotone polynomials are obtained. The estimates involve an L pmodulus of continuity or equivalently a Peetre functional that weighs differently the behavior of the function in the middle of the interval and near the end points. Recieved by the editor on September 3, 1986. Keywords and phrases: degree of monotone approximation, Jackson type estimates, L p-modulus of continuity, Peetre kernel.

Interpolatory estimates for convex piecewise polynomial approximation

Journal of Mathematical Analysis and Applications

In this paper, among other things, we show that, given r ∈ N, there is a constant c = c(r) such that if f ∈ C r [−1, 1] is convex, then there is a number N = N(f, r), depending on f and r, such that for n ≥ N, there are convex piecewise polynomials S of order r + 2 with knots at the Chebyshev partition, satisfying |f (x) − S(x)| ≤ c(r) min 1 − x 2 , n −1 1 − x 2 r ω 2 f (r) , n −1 1 − x 2 , for all x ∈ [−1, 1]. Moreover, N cannot be made independent of f. * AMS classification: 41A10, 41A25. Keywords and phrases: Convex approximation by polynomials, Degree of approximation, Jackson-type interpolatory estimates.

On 3-monotone approximation by piecewise polynomials

Journal of Approximation Theory, 2005

We consider 3-monotone approximation by piecewise polynomials with prescribed knots. A general theorem is proved, which reduces the problem of 3-monotone uniform approximation of a 3-monotone function, to convex local L 1 approximation of the derivative of the function. As the corollary we obtain Jackson-type estimates on the degree of 3-monotone approximation by piecewise polynomials with prescribed knots.

Pointwise estimates for convex polynomial approximation

Proceedings of the American Mathematical Society, 1986

For a convex function / £ C[-l, 1] we construct a sequence of convex polynomials pn of degree not exceeding n such that \f(x)-pn(x)\ < Cui2(f, vl-x2In),-1 < i < 1. If in addition / is monotone it follows that the polynomials are also monotone thus providing simultaneous monotone and convex approximation.

Interpolatory Pointwise Estimates for Polynomial Approximation X1. Introduction and Main Results

2007

We discuss whether or not it is possible to have interpolatory pointwise estimates in the approximation of a function f 2 C 0; 1], by polynomials. For the sake of completeness as well as in order to strengthen some existing results, we discuss brieey the situation in unconstrained approximation. Then we deal with positive and monotone constraints where we show exactly when such interpolatory estimates are achievable by proving aarmative results and by providing the necessary counterexamples in all other cases. The eeect of the endpoints of the nite interval on the quality of approximation of continuous functions by algebraic polynomials, was rst observed by Nikolski Nik46]. Later pointwise estimates of this phenomenon were given by Timan Tim51] (k = 1), Dzjadyk Dzj58, Dzj77] (k = 2), Freud Fre59] (k = 2), and Brudny Bru63] (k 2), who proved that if f 2 C r 0; 1], then for each n N = r + k ? 1, a polynomial p n 2 n exists, such that (1.1) jf(x) ? p n (x)j c(r; k) r n (x)! k (f (r) ; ...

Interpolatory pointwise estimates for convex polynomial approximation

Acta Mathematica Hungarica

This paper deals with approximation of smooth convex functions f on an interval by convex algebraic polynomials which interpolate f and its derivatives at the endpoints of this interval. We call such estimates "interpolatory". One important corollary of our main theorem is the following result on approximation of f ∈ ∆ (2) , the set of convex functions, from W r , the space of functions on [−1, 1] for which f (r−1) is absolutely continuous and f (r) ∞ := ess sup x∈[−1,1] |f (r) (x)| < ∞: For any f ∈ W r ∩ ∆ (2) , r ∈ N, there exists a number N = N(f, r), such that for every n ≥ N, there is an algebraic polynomial of degree ≤ n which is in ∆ (2) and such that

Uniform limits of sequences of polynomials and their derivatives

Proceedings of the American Mathematical Society, 1992

Let £ be a compact subset of the unit interval [0, 1], and let C{E) denote the space of functions continuous on E with the uniform norm. Consider the densely defined operator D: C(E)-► C(E) given by Dp = p' for all polynomials p. Let 9 represent the graph of D , that is & = {(p, p'): p polynomials} considered as a submanifold of C(E) x C(E). Write the interior of the set E , mtE as a countable union of disjoint open intervals and let E be the union of the closure of these intervals. The main result is that the closure of 3f is equal to the set of all functions (h, k) € C(E) x C(E) such that h is absolutely continuous on E and k\E = h'\E. As a consequence, the operator D is closable if and only if the set E is the closure of its interior. On the other extreme, & is dense in C(E) x C(E) i.e. for any pair (/, g) 6 C(E) x C(E), there exists a sequence of polynomials (pn} so that pn-> f and p'n-> g uniformly on E , if and only if the interior int £ of £ is empty.

Uniform convergence for sequences of best L^{p} approximation (original) (raw)

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