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1.
K. A. Kopotun 《分析论及其应用》1995,11(2):41-58
Some estimates for unconstrained and convex polynomial approximation in the uniform metric are obtained. These results are
given in terms of the Ditzian-Totik moduli of smoothness
, ≤1 with
. The construction of the approximating polynomials does not depend on λ. 相似文献
2.
Sorin G. Gal 《分析论及其应用》2002,18(1):26-33
In this paper we construct bivariate polynomials attached to a bivariate function, that approximate with Jackson-type rate involving a bivariate Ditzian-Totik ω2-modulus of smoothness and preserve some natural kinds of bivariate monotonicity and convexity of function.The result extends that in univariate case-of D. Leviatan in [5-6], improves that in bivariate case of the author in [3] and in some special cases, that in bivariate case of G. Anastassiou in [1]. 相似文献
3.
Letw be a suitable weight function,B
n,p denote the polynomial of best approximation to a functionf inL
w
p
[–1, 1],v
n
be the measure that associates a mass of 1/(n+1) with each of then+1 zeros ofB
n+1,p–B
n,p and be the arcsine measure defined by
. We estimate the rate at which the sequencev
n
converges to in the weak-* topology. In particular, our theorem applies to the zeros of monic polynomials of minimalL
w
p
norm.This author gratefully acknowledges partial support from NSA contract #A4235802 during 1992, AFSOR Grant 226113 during 1993 and The Alexander von Humboldt Foundation during both of these years. 相似文献
4.
The present paper gives a contribution of wavelet aspects to classical algebraic polynomial approximation theory. As is so often the case in classical approximation, the authors follow the pattern provided by the trigonometric polynomial case. Algebraic polynomial interpolating scaling functions and wavelets are constructed by using the interpolation properties of de la Vallée Poussin kernels with respect to the four kinds of Chebyshev weights. For the decomposition and reconstruction of a given function the structure of the involved matrices is studied in order to reduce the computational effort by means of fast discrete cosine and sine transforms.
Dedicated to Prof. Guiseppe Mastroianni on the occasion of his 65th birthday.AMS subject classification 65D05, 65T60 相似文献
5.
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
6.
A. Foulqui��?Moreno A. Mart��nez-Finkelshtein V. L. Sousa 《Constructive Approximation》2011,33(2):219-263
We consider the orthogonal polynomials on [−1,1] with respect to the weight
$w_c(x)=h(x)(1-x)^{\alpha}(1+x)^{\beta} \varXi _{c}(x),\quad\alpha,\beta>-1,$w_c(x)=h(x)(1-x)^{\alpha}(1+x)^{\beta} \varXi _{c}(x),\quad\alpha,\beta>-1, 相似文献
7.
We study properties of polynomials R n+5(x) of least deviation from zero in the L[?1, 1] metric, with five given leading coefficients whose forms were calculated previously. Theorems 1 and 2 together with Theorem A contain, in particular, a final classification of polynomials R n+5(x) that have exactly (n + 1) sign changes in (?1, 1). 相似文献
8.
9.
In this paper we calculate the upper bounds of the best one-sided approximations, by trigonometric polynomials and splines of minimal defect in the metric of the space L, of the classes WrH (r = 2, 4, 6, ...) of all 2-periodic functions f(x) that are continuous together with their r-th derivative fr(x) and such that for any points x and x we have ¦f
r (x) fr (x) ¦ (x–x¦), where (t) is a modulus of continuity that is convex upwards.Translated from Matematicheskie Zametki, Vol. 21, No. 3, 313–327, March, 1977. 相似文献
10.
11.
Adam P. Wójcik 《Monatshefte für Mathematik》1988,105(1):75-81
LetE be a compact subset of the complex planeC such that Leja's extremal functionL
E
forE is continuous. If almost all zeros of the polynomials of best approximation to a functionfC(E) are outside the setE
R
={zC:L
E
(z<R)}, for someR>1, thenf is extendible to a holomorphic function inE
R
. If the zeros ofn-th, polynomial of best approximation tof are outside
and the sequence {R
n
–n
} rapidly decreases to zero thenf can be extended to aC
function on 075-4}. 相似文献
12.
Chebyshev determined $$\mathop {\min }\limits_{(a)} \mathop {\max }\limits_{ - 1 \le x \le 1} |x^n + a_1 x^{n - 1} + \cdots + a_n |$$ as 21?n , which is attained when the polynomial is 21?n T n(x), whereT n(x) = cos(n arc cosx). Zolotarev's First Problem is to determine $$\mathop {\min }\limits_{(a)} \mathop {\max }\limits_{ - 1 \le x \le 1} |x^n - n\sigma x^{n - 1} + a_2 x^{n - 2} + \cdots + a_n |$$ as a function ofn and the parameter σ and to find the extremal polynomials. He solved this in 1878. Another discussion was given by Achieser in 1928, and another by Erdös and Szegö in 1942. The case when 0≤|σ|≤ tan2(π/2n) is quite simple, but that for |σ|> tan2(π/2n) is quite different and very complicated. We give two new versions of the proof and discuss the change in character of the solution. Both make use of the Equal Ripple Theorem. 相似文献
13.
Let be a positive number, and letE
n,n
(x
;[0,1]) denote the error of best uniform rational approximation from
n,n
tox
on the interval [0,1]. We rigorously determined the numbers {E
n,n
(x
;[0,1])}
n
=1/30
for six values of in the interval (0, 1), where these numbers were calculated with a precision of at least 200 significant digits. For each of these six values of , Richardson's extrapolation was applied to the products
to obtain estimates of
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