共查询到20条相似文献,搜索用时 9 毫秒
1.
G. A. Dzyubenko 《Ukrainian Mathematical Journal》1996,48(3):367-376
We prove that if a functionf ∈C (1) (I),I: = [?1, 1], changes its signs times (s ∈ ?) within the intervalI, then, for everyn > C, whereC is a constant which depends only on the set of points at which the function changes its sign, andk ∈ ?, there exists an algebraic polynomialP n =P n (x) of degree ≤n which locally inherits the sign off(x) and satisfies the inequality $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$ , where ω k (f′;t) is thekth modulus of continuity of the functionf’. It is also shown that iff ∈C (I) andf(x) ≥ 0,x ∈I then, for anyn ≥k ? 1, there exists a polynomialP n =P n (x) of degree ≤n such thatP n (x) ≥ 0,x ∈I, and |f(x) ?P n (x)| ≤c(k)ω k (f;n ?2 +n ?1 √1 ?x 2),x ∈I. 相似文献
2.
Zhijian Qiu 《数学学报(英文版)》2009,25(7):1217-1222
Let G be an open subset in the extended complex plane and let A(G) denote the algebra of all functions analytic on G and continuous on G. We call a domain multi-nicely connected if there is a circular domain W and a conformal map ~ from W onto G such that the boundary value function of φ is univalent almost everywhere with respect to the arclength on aW. Suppose that every component of G is finitely connected and none of the components of G have single point boundary components. We show that for every bounded analytic function on G to be the pointwise limit of a bounded sequence of functions in A(G), it is necessary and sufficient that each component of G is multi-nicely connected and the harmonic measures of G are mutually singular. This generalizes the corresponding result of Davie for the case when the components of G are simply connected. 相似文献
3.
We estimate the rate of covergence to functions in the spacesL
p [0,1] and C[0,1] by polynomial of the form ∑
λ
α
λ
x
λ, where the λ′s are positive real numbers and 0. 相似文献
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A. A. Gonchar 《Mathematical Notes》1972,11(2):83-86
For best piecewise polynomial approximation n=n (f; [0, 1]) of a functionf, which is continuous on the interval [0, 1] and admits a bounded analytic continuation onto the disk K=z:¦z–1¦<, the relation n=o[
f
(e
–n
)] is valid.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 129–134, February, 1972. 相似文献
6.
YEMAODONG 《高校应用数学学报(英文版)》1998,13(1):117-122
In this paper a representation for linear polynomial projection is given. Then it isshown that the near minimax approximation is equivalant to a problem of the best approxima-tion of hivariste functions. And some properties for near minimax are obtained. 相似文献
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Weighted exponential polynomial approximation 总被引:7,自引:0,他引:7
邓冠铁 《中国科学A辑(英文版)》2003,46(2):280-287
A necessary and sufficient condition for completeness of systems of exponentials with a weight in Lp is established and a quantitative relation between the weight and the system of exponential in Lp is obtained by using a generalization of Malliavin's uniqueness theorem about Watson's problem. 相似文献
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A. F. Kalaida 《Journal of Mathematical Sciences》1994,69(5):1280-1284
Conditions are established when the collocation polynomials Pm(x) and PM(x), m M, constructed respectively using the system of nodes xj of multiplicities aj 1, j = O,, n, and the system of nodes x-r,,xo,,xn,,xn+r1, r O, r1 O, of multiplicities a-r,,(ao + yo),,(an + yn),,an+r1, aj + yj 1, are two sided-approximations of the function f on the intervals
, xj[, j = O,...,n + 1, and on unions of any number of these intervals. In this case, the polynomials Pm (x), PM
(l) (x) with l aj are two-sided approximations of the function f(1) in the neighborhood of the node xj and the integrals of the polynomials Pm(x), PM(x) over Dj are two-sided approximations of the integral of the function f (over Dj). If the multiplicities aj aj + yj of the nodes xj are even, then this is also true for integrals over the set j=
µ
k
Dj µ 1, k n. It is shown that noncollocation polynomials (Fourier polynomials, etc.) do not have these properties.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 31–37, 1989. 相似文献
12.
In the paper, the uniform approximation of a circle arc (or a whole circle) by a parametric polynomial curve is considered. The approximant is obtained in a closed form. It depends on a parameter that should satisfy a particular equation, and it takes only a couple of tangent method steps to compute it. For low degree curves, the parameter is provided exactly. The distance between a circle arc and its approximant asymptotically decreases faster than exponentially as a function of polynomial degree. Additionally, it is shown that the approximant could be applied for a fast evaluation of trigonometric functions too. 相似文献
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Asymptotic approximation with Kantorovich polynomial 总被引:5,自引:1,他引:4
Ulrich Abel 《分析论及其应用》1998,14(3):106-116
We present the complete asymptotic expansion for the Kantorovich polyno-mials K_n as n→∞.The result is in a form convenient for applications.All coefficients of n~(-k)(k=1,2,…)are calculated explicitly in terms ofStifling numbers of the first and second kind.Moreover,we treat the simultaneous approximation with Kantorovich poly-nomials and determine the rate of convergence of d/(dx)K_n(f;x)-f'(x). 相似文献
16.
The use of successive polynomial spline approximation is established as a method of improving the accuracy of estimates of derivatives of periodic functions approximated by interpolating odd order splines defined on a uniformly spaced set of data points. For the various configurations possible with this multiple-approximation method, bounds for the leading error terms are explicitly given. In particular, for the quintic spline, the variety of approximation sequences is described in detail. 相似文献
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Charles B. Dunham 《BIT Numerical Mathematics》1974,14(2):133-135
The quality of a polynomial approximation on an interval to a functionf is considered as a function of its points of interpolation. Iff satisfies a Lipschitz condition of order 1, the quality depends linearly on the distance of the points of interpolation from an optimal interpolating point set: further restrictions onf still give only linear dependence. This suggests that algorithms based on interpolation are inferior to algorithms based on error extrema (such as the Remes algorithm). 相似文献
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We show that ifw(x)=exp(–|x|), then in the case =1 for every continuousf that vanishes outside the support of the corresponding extremal measure there are polynomialsP
n of degree at mostn such thatw
n
P
n uniformly tends tof, and this is not true when <1. these=" are=" the=" missing=" cases=" concerning=" approximation=" by=" weighted=" polynomials=" of=" the=">1.>w
n
P
n wherew is a Freud weight. Our second theorem shows that even if we are only interested in approximation off on the extremal support, the functionf must still vanish at the endpoints, and we actually determine the (sequence of) largest possible intervals where approximation is possible. We also briefly discuss approximation by weighted polynomials of the formW(anx)P
n
(x).Communicated by Edward B. Saff. 相似文献