Monotone and comonotone approximation

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.

Journal of Approximation Theory, 1998

Let f ∈ C[−1, 1] change its convexity finitely many times, in the interval. We are interested in estimating the degree of approximation of f by polynomials which are coconvex with it, namely, polynomials that change their convexity exactly at the points where f does. We discuss some Jackson type estimates where the constants involved depend on the location of the points of change of convexity. We also show that in some cases the constants may be taken independent of the points of change of convexity, but that in other cases this dependence is essential. But mostly we obtain such estimates for functions f that themselves are continuous piecewise polynomials on the Chebyshev partition, which form a single polynomial in a small neighborhood of each point of change of convexity. These estimates involve the k modulus of smoothness of the piecewise polynomials when they themselves are of degree k − 1.

1996

When we approximate a continuous nondecreasing function f in ?1;1], we wish sometimes that also the approximating polynomials be nondecreasing. However, this constraint restricts very much the degree of approximation that the polynomials can achieve, namely, only the rate of ! 2 (f; 1=n). It turns out as we will prove somewhere else that relaxing the monotonicity requirement in intervals of length 1=n 2 near the endpoints allows the polynomials to achieve the rate of ! 3 . On the other hand, we show in this paper, that even when we relax the requirement of monotonicity of the polynomials on sets of measures approaching 0, (no matter how slowly or how fast), ! 4 is not reachable.

Proceedings of the American Mathematical Society, 1983

Jackson type theorems are established for the approximation of a function / that changes sign finitely many times in [-1,1] by polynomials p" which are copositive with it (i.e. fp" > 0 on [-1,1]). The results yield the rate of nonconstrained approximation and are thus best possible in the same sense as in the nonconstrained case.

Journal of Approximation Theory, 1974

Let P, be the set of all algebraic polynomials of degree II or less. For f~ C[u, b], the degree of approximation to f by polynomials in P, is En(f) = inf{llf-p 11: p E PJ, where the norm is the uniform norm. Jackson's theorem [l] states that there exists C > 0 such that En(f) < Cw(f; l/n), where w(f; 6) is the modulus of continuity off. f is said to be piecewise monotone if it has only a finite number of local maxima and minima in [a, b]. The local maxima and minima in (a, b) are called the peaks off. Wolibner [5] has shown that for any E > 0 there exists a polynomial, p, such that IIfp II < E and p is comonotone with f; i.e., p increases and decreases simultaneously with $ Let E,*(f) = inf{llf-p 11: p E P, , p comonotone with f}. Clearly E,*cf) 3 E,df). We seek an upper bound on E,*(f). Forfmonotone, Lorentz and Zeller [3] have shown that there exists C, > 0 such that E,*(f) < C,w(f; I/n). Newman, Passow, and Raymon [4] have obtained results of a modified nature. They have shown that there exists p E P, satisfying IIfp II < C+(f; l/n), C, an absolute constant, such that f and p are comonotone except in certain neighborhoods (whose diameters tend to zero with n) of the peaks. In this note we obtain a comonotone approximation on the entire interval [a, b], but at a sacrifice in the accuracy of approximation. LEMMA 1. Let fe C(j+l)[a, b] and suppose that f(u) = 0, u E (a, b). Let g(x) =f[x, u], the divided dzfirence off, where we dejine f[u, U] =f'(u). Then g E Cj[u, b] and II g(j) 11 < (j + 1)-l /lf(j+') 11.

. When we approximate a continuous function f which changes its monotonicity finitely many, say s times, in [Gamma1; 1], we wish sometimes that the approximating polynomials follow these changes in monotonicity. However, it is well known that this requirement restricts very much the degree of approximation that the polynomials can achieve, namely, only the rate of ! 2 (f; 1=n) and even this not with a constant (dependent only on s), rather with a constant which depends on the location of the interior extrema. Recently, we proved that relaxing the comonotonicity requirement in intervals of length proportional to 1=n about the interior extrema of the function and in intervals of length 1=n 2 near the endpoints, what we called nearly comonotone approximation, allows the polynomials to achieve a pointwise approximation rate of ! 3 (moreover, with a constant which depends only on s). We show here that even when we relax the requirement of monotonicity of the polynomials on sets of measur...

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) ; ...

New Developments in Approximation Theory, 1999

We survey the Jackson and Jackson type estimates for comonotone polynomial approximation of continuous and r-times continuously di erentiable functions which change their monotonicity nitely many times in a nite interval, say ?1;1], with special attention to the constants involved in the estimates. We describe four possibilities ranging from the existence of estimates involving absolute constants and which are valid for polynomials of all degrees except the very few rst ones, through estimates where either the constants or the degrees of the polynomials depend on the location of the points of change of monotonicity, then through estimates where either the constants or the degrees of the polynomials depend on the speci c function, and nally cases where there are no Jackson type estimates possible.

1999

Abstract. Let f ∈ C[−1, 1] change its convexity finitely many times, in the interval. We are interested in estimating the degree of approximation of f by polynomials, and by piecewise polynomials, which are coconvex with it, namely, polynomials and piecewise polynomials that change their convexity exactly at the points where f does. We obtain Jackson type estimates and summarize the positive and negative results in a truth-table as we have previously done for comonotone approximation. Let f ∈ C[−1, 1] change its convexity finitely many times, say s ≥ 0 times, in the interval. We are interested in estimating the degree of approximation of f by polynomials which are coconvex with it, namely, polynomials that change their convexity exactly at the points where f does.

Journal of Approximation Theory, 1982