Unconstrained and convex polynomial approximation inC[−1,1]

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.     

Polynomials of best approximation inC[−1,1]

Equivalences between the condition |P _n ^(k) (x)|≦K(n ⁻¹√1−x ²+1/n ²) ^k n ^-a, whereP _n(x) is the bestn-th degree polynomial approximation tof(x), and the Peetre interpolation space betweenC[−1,1] and the space (1−x ²) ^k f ^(2k)(x)∈C[−1,1] is established. A similar result is shown forE _n(f)= ‖f−P _n‖ _C[−1,1]. Rates other thann ^-a are also discussed. Supported by NSERC grant A4816 of Canada.     

Interpolating polynomial wavelets on [−1,1]

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     

Convex polynomial and spline approximation in C[−1, 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.     

SHAPE-PRESERVING BIVARIATE POLYNOMIAL APPROXIMATION IN C([-1,1]×[-1,1])

In thit paper we construct bivariate polynomials attached to a btoariate function, that approximate with Jackson-type rate involving a btoariate Ditzian-Tatik ωξmodulus 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].     

A discrepancy theorem concerning polynomials of best approximation inL w p [−1, 1]

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.     

A new central polynomial for 3 × 3 matrices

Abstract

Orthodox semigroups have been studied by many authors, in particular by Hall, Yamada and Petrich. In this paper, we give the standard representation of orthodox semigroups and investigate various e-varieties of orthodox semigroups which are determined by the standard representations.     

Convex polynomial and spline approximation inL p,O<p<∞

We prove that a convex functionf ∈ L _p[−1, 1], 0<p<∞, can be approximated by convex polynomials with an error not exceeding Cω ₃ ^ϕ (f,1/n)_p where ω ₃ ^ϕ (f,·) is the Ditzian-Totik modulus of smoothness of order three off. We are thus filling the gap between previously known estimates involving ω ₃ ^ϕ (f,1/n)_p, and the impossibility of having such estimates involving ω₄. We also give similar estimates for the approximation off by convexC ⁰ andC ¹ piecewise quadratics as well as convexC ² piecewise cubic polynomials. Communicated by Dietrich Braess     

On the finite element approximation for maxwell’s problem in polynomial domains of the plane

The time-harmonic Maxwell boundary value problem in polygonal domains of R² is considered. The behaviour of the solution in the neighbourhood of nonregular boundary points is given and asymptotic error estimates in L₂- and in curl-div-norm for a finite element approximation of the solution are derived     

Are the degrees of best (co)convex and unconstrained polynomial approximation the same?

Let ℂ[−1,1] be the space of continuous functions on [−,1], and denote by Δ² the set of convex functions f ∈ ℂ[−,1]. Also, let E _n (f) and E _n ⁽²⁾ (f) denote the degrees of best unconstrained and convex approximation of f ∈ Δ² by algebraic polynomials of degree < n, respectively. Clearly, En (f) ≦ E _n ⁽²⁾ (f), and Lorentz and Zeller proved that the inverse inequality E _n ⁽²⁾ (f) ≦ cE _n (f) is invalid even with the constant c = c(f) which depends on the function f ∈ Δ². In this paper we prove, for every α > 0 and function f ∈ Δ², that

The limit spectral graph in semiclassical approximation for the Sturm–Liouville problem with complex polynomial potential

The limit behavior of the discrete spectrum of the Sturm–Liouville problem whose potential is a polynomial with complex coefficients on an interval, on a half-axis, and on the entire axis is studied. It is shown that, at large parameter values, the eigenvalues are concentrated along the so-called limit spectral graph; the curves forming this graph are classified. Asymptotics of eigenvalues along curves of various types in the graph are calculated.     

Zolotarev's first problem — the best approximation by polynomials of degree ≤n − 2 tox n − nσx n−1 in [−1, 1]

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 2^1?n , which is attained when the polynomial is 2^1?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.     

General approximation schemes for min–max (regret) versions of some (pseudo-)polynomial problems

While the complexity of min–max and min–max regret versions of most classical combinatorial optimization problems has been thoroughly investigated, there are very few studies about their approximation. For a bounded number of scenarios, we establish general approximation schemes which can be used for min–max and min–max regret versions of some polynomial or pseudo-polynomial problems. Applying these schemes to shortest path, minimum spanning tree, minimum weighted perfect matching on planar graphs, and knapsack problems, we obtain fully polynomial-time approximation schemes with better running times than the ones previously presented in the literature.     

L_([-1,1])~p(1

梅雪峰  周颂平 《数学年刊A辑(中文版)》2004,(1)

Polynomials of least deviation from zero in the L[−1, 1] metric with five fixed coefficients

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).