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1.
We prove that a convex functionf C[–1, 1] can be approximated by convex polynomialsp
n
of degreen at the rate of 3(f, 1/n). We show this by proving that the error in approximatingf by C2 convex cubic splines withn knots is bounded by 3(f, 1/n) and that such a spline approximant has anL
third derivative which is bounded by n33(f, 1/n). Also we prove that iff C2[–1, 1], then it is approximable at the rate ofn
–2 (f, 1/n) and the two estimates yield the desired result.Communicated by Ronald A. DeVore. 相似文献
2.
В. В. Арестов 《Analysis Mathematica》1991,17(1):11-20
相似文献
3.
D. S. Lubinsky 《Constructive Approximation》1991,7(1):501-519
We show that the most of the time, most poles of diagonal multipoint Padé or best rational approximants to functions admitting fast rational approximation, leave the region of meromorphy. Following is a typical result: Letf be single-valued and analytic in CS, where cap(S)=0. Let {n
j
}
j=1
be an increasing sequence of positive integers withn
j+1/n
j
1 asj. Then there exists an infinite sequenceL of positive integers such that asj,jL the total multiplicity of poles of any sequence of type (n
j
,n
j
) multipoint Padé or best rational approximants tof, iso(n
j
) in any compactK in whichf is meromorphic. The sequenceL is independent of the particular sequence of multipoint Padé or best approximants, and yields the same behavior for near-best approximants. If the errors of best approximation on some compact set satisfy a weak regularity condition, then we may takeL={1,2,3,}.Communicated by Edward B. Saff. 相似文献
4.
E. Tarafdar 《Monatshefte für Mathematik》1976,53(1):341-344
Iff is a self mapping on a closed convex subsetK of a separated quasicomplete locally convex linear topological spaceE such that (i)E is strictly convex, (ii)f (K) is contained in a compact subset ofK and (iii)f satisfies a contraction condition, then it is shown that for eachxK, the sequence of {U
n
(x)}
n
=1 of iterates, whereU
KK is defined byU
(y)=f(y)+(1-) y, yK, converges to a fixed point off. 相似文献
5.
Andreas Schief 《Monatshefte für Mathematik》1989,108(1):59-70
LetX andY be Hausdorff spaces and denote byM (X) andM (Y) the corresponding spaces of finite and non-negative Borel measures, endowed with the weak topology. A Borel map :XY induces the map
:M (X)M (Y). We give necessary and sufficient conditions for
to be open. In case of being a surjection between Suslin spaces,
is open if and only if is. 相似文献
6.
Let C be a simply connected domain, 0, and let n,nN, be the set of all polynomials of degree at mostn. By n() we denote the subset of polynomials p n withp(0)=0 andp(D), whereD stands for the unit disk {z: |z|<1}, and=" by=">1},>we denote the maximal range of these polynomials. Letf be a conformal mapping fromD onto ,f(0)=0. The main theme of this note is to relate n (or some important aspects of it) to the imagesf
s
(D), wheref
s
(z):=f[(1–s)z], 0s<1. for=" instance=" we=" prove=" the=" existence=" of=" a=" universal=">1.>c
0 such that, forn2c
0, 相似文献
7.
A 2-periodic continuous real functionf is said to beperiodically monotone if it has the following property: there exist numbert
1t
2t
3t
1+2 such thatf is nonincreasing fort
1t
2 and nondecreasing int
2tt
3. For any 2-periodic, integrable real functiong with
0
2
|g(t|dt) we define
|