Piecewise Monotone Pointwise Approximation

Section: Jackson-type Propositionssupporting

confidence: 52%

Piecewise Coapproximation and the Whitney Inequality

Pleshakov

Shatalina

2000

“…We begin by quoting some results from [7] (see Lemma 5.3 there and the definitions above it). We put b :=max(6s, 8k&s+1), and for each j such that I j & O=< (which we denote by j # H), we let the polynomials T j (x)=T j, n (x; b; Y ) and T j (x) :=T j, n (x; b; Y ) of degree (b+1)(4n&2)+s+2 be those defined there.…”

Section: )mentioning

confidence: 99%

“…Let x &1 :=1, x n+1 := &1 and for each j=0, ..., n, set x j :=x j, n := cos( j?Ân), I j :=I j, n :=[x j , x j&1 ], and h j :=h j, n := |I j | :=x j&1 &x j . For later reference we need the following well known relations (see, e.g., [7]) \ n (x)<h j <5\ n (x),…”

Section: âN+1ânmentioning

confidence: 99%

Nearly Comonotone Approximation

Leviatan

Shevchuk

1998

We discuss the degree of approximation by polynomials of a function f that is piecewise monotone in [&1, 1]. We would like to approximate f by polynomials which are comonotone with it. We show that by relaxing the requirement for comonotonicity in small neighborhoods of the points where changes in monotonicity occur and near the endpoints, we can achieve a higher degree of approximation. We show here that in that case the polynomials can achieve the rate of | 3 . On the other hand, we show in another paper, that no relaxing of the monotonicity requirements on sets of measures approaching 0 allows | 4 estimates. Academic Press

“…. ; c 5 and C 0 ; when we have a reason to keep track of them in the computations that we have to carry out in the proofs.…”

Section: Introductionmentioning

confidence: 99%

Coconvex Approximation

Leviatan

Shevchuk

2002

Let f 2 C½À1; 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 À 1: # 2002 Elsevier Science (USA)