三角多项式类中的Birkhoff型三角插值

本文在三角多项式类中讨论了2π周期函数的一类Birkhoff型等距结点的三角插值问题,给出了此问题有解的充要条件,并构造出插值基.     

插值三角多项式在Orlicz空间逼近的渐进等式

对于具有等距分布插值结点的三角多项式,借助广义的Minkowski不等式在Orlicz空间内建立了由三角多项式逼近的渐近等式.并对于Orlicz空间内不同的函数类给出不同的结果.     

On the Decay of Infinite Products of Trigonometric Polynomials

We consider infinite products of the form f(=_k=1 m _k(2^-k), where {m _k} is an arbitrary sequence of trigonometric polynomials of degree at most n with uniformly bounded norms such that m _k(0)= 1 for all k. We show that f() can decrease at infinity not faster than O(^-n) and present conditions under which this maximal decay is attained. This result can be applied to the theory of nonstationary wavelets and nonstationary subdivision schemes. In particular, it restricts the smoothness of nonstationary wavelets by the length of their support. This also generalizes well-known similar results obtained for stable sequences of polynomials (when all m _k coincide). By means of several examples, we show that by weakening the boundedness conditions one can achieve exponential decay.     

Lacunary Interpolation by Antiperiodic Trigonometric Polynomials

The problem of lacunary trigonometric interpolation is investigated. Does a trigonometric polynomial T exist which satisfies T(x _k) = a _k, D ^m T(x _k) = b _k, 0 k n – 1, where x _k = k/n is a nodal set, a _k and b _k are prescribed complex numbers, and m N. Results obtained by several authors for the periodic case are extended to the antiperiodic case. In particular solvability is established when n as well as m are even. In this case a periodic solution does not exist.     

A Sharpening of the Taikov Lower Bound in the Inequality between the C- and L-Norms for Trigonometric Polynomials

Mathematical Notes -     

关于一类新的Bernstein型三角求和多项式

A new family of trigonometric summation polynomials, Gn,r(f;θ), of Bernstein type is constructed. In contrast to other trigonometric summation polynomials, the convergence properties of the new polynomials are superior to others. It is proved that Gn,r(f;θ) converges to arbitrary continuous functions with period 2π uniformly on (-∞, ∞) as n→∞. In particular, Gn,r(f;θ) has the best convergence order, and its saturation order is 1/n2r 4.     

On Extremal Properties of Interpolation Polynomials for Functions of Several Variables

By using earlier results, we find solutions of two extremal problems on sets of interpolation polynomials for functions of several variables.     

A Class of Extremal Functions and Trigonometric Polynomials

In this paper we present two classes of extremal approximating functions. These functions have the property that they are entire, have finite exponential type, and provide excellent approximations along the real line for a specific set of functions. One class of functions provides majorants and minorants, while the other class minimizes theL¹-norm on the real line. As applications we construct extremal trigonometric polynomials and obtain an inequality involving almost periodic trigonometric polynomials.     

Interpolation Formula for Functions of Several Variables

Mathematical Notes -     

Trigonometric Convex Underestimator for the Base Functions in Fourier Space

A three-parameter (a, b, x_s) convex underestimator of the functional form (x) = -a sin[k(x-x_s)] + b for the function f(x) = sin(x+s), x [x_L, x_U], is presented. The proposed method is deterministic and guarantees the existence of at least one convex underestimator of this functional form. We show that, at small k, the method approaches an asymptotic solution. We show that the maximum separation distance of the underestimator from the minimum of the function grows linearly with the domain size. The method can be applied to trigonometric polynomial functions of arbitrary dimensionality and arbitrary degree. We illustrate the features of the new trigonometric underestimator with numerical examples.Support from the National Science Foundation and the National Institutes of Health Grant R01 GM52032 is gratefully acknowledged.     

On Polynomials over a Finite Field of Even Characteristic with Maximum Absolute Value of the Trigonometric Sum

We study trigonometric sums in finite fields . The Weil estimate of such sums is well known: , where f is a polynomial with coefficients from F(Q). We construct two classes of polynomials f, , for which attains the largest possible value and, in particular, .     

Positive Values of Polynomials

Mathematical Notes -     

关于两个多元多项式互素问题

给出了两个二元多项式互素的充要条件 ,然后利用这个充要条件推出二元多项式互素的性质 ,最后给出一般的 n元多项式互素的充要条件 .     

Nonnegative Trigonometric Polynomials

An extremal problem for the coefficients of sine polynomials, which are nonnegative in [0,π] , posed and discussed by Rogosinski and Szegő is under consideration. An analog of the Fejér—Riesz representation of nonnegative general trigonometric and cosine polynomials is proved for nonnegative sine polynomials. Various extremal sine polynomials for the problem of Rogosinski and Szegő are obtained explicitly. Associated cosine polynomials k _n (θ) are constructed in such a way that { k _n (θ) } are summability kernels. Thus, the L _p , pointwise and almost everywhere convergence of the corresponding convolutions, is established. April 26, 2000. Date revised: December 28, 2000. Date accepted: February 8, 2001.     

Integral Norms of Trigonometric Polynomials

For trigonometric polynomials with coefficients equal to 1 or 0 in absolute value whose spectra are located on the left-hand side of binary blocks, we establish two-sided estimates of the L ₁-norm.     

一类三角插值多项式在Besov空间中的逼近

利用K泛函的定义首次研究了在Besov空间中,一类三角插值多项式的逼近和饱和问题,确定了逼近的饱和类与饱和阶.     

Orlicz空间中三角多项式倒数对周期可微函数的逼近定理

利用Orlicz空间内有关不等式技巧在Orlicz空间内研究了用三角多项式的倒数逼近周期可微函数的问题.得到了一个逼近定理及其推论.     

Characteristic and Ehrhart Polynomials

Let A be a subspace arrangement and let (A,t) be the characteristic polynomial of its intersection lattice L( A). We show that if the subspaces in A are taken from , where is the type B Weyl arrangement, then (A,t) counts a certain set of lattice points. One can use this result to study the partial factorization of (A,t) over the integers and the coefficients of its expansion in various bases for the polynomial ring R[t]. Next we prove that the characteristic polynomial of any Weyl hyperplane arrangement can be expressed in terms of an Ehrhart quasi-polynomial for its affine Weyl chamber. Note that our first result deals with all subspace arrangements embedded in while the second deals with all finite Weyl groups but only their hyperplane arrangements.     

Intractability Results for Positive Quadrature Formulas and Extremal Problems for Trigonometric Polynomials

Lower bounds for the error of quadrature formulas with positive weights are proved. We get intractability results for quasi-Monte Carlo methods and, more generally, for positive formulas. We consider general classes of functions but concentrate on lower bounds for relatively small classes of trigonometric polynomials. We also conjecture that similar lower bounds hold for arbitrary quadrature formulas and state different equivalent conjectures concerning positive definiteness of certain matrices and certain extremal problems for trigonometric polynomials. We also study classes of functions with weighted norms where some variables are “more important” than others. Positive quadrature formulas are then tractable iff the sum of the weights is bounded.     

新二元三角插值多项式的构造

设Ω=[-πxπ,-πyπ],C(Ω)表示关于x,y均以2π为周期的连续函数空间.若f(x,y)∈C(Ω),取结点组为(xk,yl)=(2k+2n 1)π,(2l 2+m 1)πk=0,1,2,…,2n,l=0,1,2,…,2m,则我们获得一个二元三角插值多项式Cn,m(f;x,y)=M1N∑k=2n0∑l=2m0f(xk,yl).1+2∑nα=1cosα(x-xk)+2∑mβ=1cosβ(y-yl)+4∑nα=1∑mβ=1cosα(x-xk)cosβ(y-yl)其中M=2m+1,N=2n+1.为改进其收敛性,本文构造一个新的因子ρα,β,使得带有该因子ρα,β的二元三角插值多项式Ln,m(f;x,y)可以在全平面上一致地收敛到每个连续的f(x,y),且具有最佳逼近阶.