实轴上Hermite插值算子的收敛性

The present paper investigates the convergence of Hermite interpolation operators on the real line.The main result is: Given 0 <δ0 < 1/2,0 < ε0 < 1.Let f ∈ C(-∞,∞) satisfy |yk| = O(e(1/2-δ0)x2k) and |f(x)| = O(e(1-ε0)x2).Then for any given point x ∈R,we have limn→∞ Hn(f,x) = f(x).     

ASYMPTOTIC REPRESENTATION FOR REMAINDER OF QUASI-HERMITE-FEJER INTERPOLATION POLYNOMIAL

Let Q_(2n+1)(f,x)be the quasi-Hermite-Fejer interpolation polynomial of functionf(x)∈C_[-1,1]based on the zeros of the Chebyshev polynomial of the second kind U_n(x)=sin((n+l)arccosx)/sin(arc cosx). In this paper, the uniform asymptotic representation for thequantity| Q_(2n+l)(f, x) -f(x) |is given. A similar result for the Hermite-Fejer interpolationpolynomial based on the zeros of the Chebyshev polynomial of the first kind is alsoestablished.     

(0,1, …, m - 2, m) INTERPOLATION FOR THE LAGUERREABSCISSAS

A necessary and sufficient condition of regularity of (0,1,…, m - 2, m) interpo-lation on the zeros of the Laguerre polynomials Ln(α) (x) (α≥ -1) in a manageable form is established. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, it is shown that, if the problem of (0,1,…, m - 2, TO) interpolation has an infinity of solutions, then the general form of the solutions is f0(x) Cf1(x) with an arbitrary constant C.     

Lagrange插值逼近导数的平均收敛

<正>We consider the rate of mean convergence of derivatives by Lagrange interpolation operators L_n(f,x) based on the zeros of Chebyshev polynomials of the first kind.A sharp estimate of the derivative of L_n(f,x)—f(x) in terms of the error of best approximation by polynomials of degree n is derived.     

On approximation of smooth functions from null spaces of optimal linear differential operators with constant coefficients

For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln(ψ) = n ∑ k=0 akψ(k), where the constant coefficients ak ∈ R may be adapted to f . We prove that for each f ∈ C(n)(I), there is a selection of coefficients {a1, ,an} and a corresponding linear combination Sn( f ,t) = n ∑ k=1 bkeλkt of functions ψk(t) = eλkt in the nullity of L which satisfies the following Jackson’s type inequality: f (m) Sn(m )( f ,t) ∞≤ |an|2n|Im|1/1q/ep|λ|λn|n|I||nm1 Ln( f ) p, where |λn| = mka x|λk|, 0 ≤ m ≤ n 1, p,q ≥ 1, and 1p + q1 = 1. For the particular operator Mn(f) = f + 1/(2n) f(2n) the rate of approximation by the eigenvalues of Mn for non-periodic analytic functions on intervals of restricted length is established to be exponential. Applications in algorithms and numerical examples are discussed.     

On the Schinzel identity ofCyclotomic Polynomial

For integer n > 0, let n.(x) denote the nth cyclotomic polynomialwhere is a primitive nib root of unity and (j, n) denotes the greatest common divisor of j andn.Although 4.(x) is irreducible over the integers, 4.(x) may be reducible over certain quadraticfield. Let n > 1 be an odd square-free number. Aurifeuille and Le Lasseur[1] proved thatLater on Schinzel[2] proved that (1) can be improved aswhere m|n,(3) denotes the Jacobi symbol, andwhere both Pn,m(x) and Qn,m(x) are polynomials with i…     

An Extension of the Turan Inequality in L_p-space for 0

周颂平 《数学研究与评论》1986,(2)

A Remark on the Generalized Roper-Suffridge Extension Operator for Spirallike Mappings of Type β and Order α

Suppose f is a spirallike function of type β and order α on the unit disk D.let Ωn,p1,p2,…,pn={z=(z1,z2,…,zn)'∈Cn:nΣj=1|zj|pj<1},where 1≤p1≤2,pj≥1,j=2,…,n,are real numbers. In this paper,we will prove that Фn,β2,у2…βn,уn(f)(z)=(f(z1),(f(z1)z1)β2(f'(z1))у2z2,…,(f(z1)z1)βn(f'(z1))уnzn)' Preserves spirallikeness of type β and order α on Ωn,p1,p2,…pn.     

ON SIMULTANEOUS APPROXIMATION TO A DIFFERENTIABLE FUNCTION AND ITS DERIVATIVE BY INVERSE PAL-TYPE INTERPOLATION POLYNOMIALS

Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P'n-1(x). By the theory of the inverse Pal-Type interpolation, for a function f(x) ∈ C[-1 1], there exists a unique polynomial Rn(x) of degree 2n - 2 (if n is even) satisfying conditions Rn(f,ξk) = f(∈ek)(1≤ k≤ n - 1) ;R'n(f,xk) = f'(xk)(1≤ k≤ n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {Rn(f,x)} (n is even) and the main result of this paper is that if f ∈ C'[1,1], r≥2, n≥ + 2> and n is even thenholds uniformly for all x ∈ [- 1,1], where h(x) = 1 +     

ON A SET OF h-HARMONIC HOMOGENEOUS POLYNOMIALS

In this paper, we study the homogeneous polynomials orthogonal with the weight function h(x(d))=x1^2k1…xd^2kd on S^(d-1).We obtain the explicit formula on a basis of the orthogonal homogeneous polynomials of degree n. It is simpler than the formula in,and can be regarded as an extension of under the weighted case.     

ON CONVERGENCE OF PAL-TYPE INTERPOLATION POLYNOMIALS

Let {x_k~*}_(k=1)~(n-1) be the zeros of the (n-1) -th Legendre polynomial p_(n-1)(x) and {x_k}_(a=1)~n be the zeros of the polynomial w(x)= (1-x2~)p_(n-1)~1(x). By the theory of the Pal interpolation, for afunction f ∈ C_([-1,1])~1, there exists a unique polynomial Q_n(f, x) of degree 2n-1 satisfying conditions Q_n(f, x_k)=f(x_k), Q'_n(f, x_k~*)=f'(x_k~*), where k=1, 2, …, n and x_n~*=-1. The main result of this paper is that if f ∈ C_([-1,1])~r, thenf(x)-Q_n(f, x)=O(1)W(x)w(f~(r), 1/n)n~((1/2)-r), -1≤x≤1.Hence, if f ∈ C_[-1,1])~1, then Q_n(f, x) converges to the function f(x)uniformly on the interval [-1, 1].     

Some Results Concerning Growth of Polynomials

Let P(z) be a polynomial of degree n having no zeros in |z|< 1, then for every real or complex number β with |β|≤ 1, and |z|=1, R ≥ 1, it is proved by Dewan et al. [4] that ︱P(Rz)+ β( R+1/2 )n P(z)︱≤ 1 /2 { (︱Rn + β(R+1/2 )n︱+︱1+ β (R + 1 /2 )n︱) max |z|=1 |P(z)︱-(︱Rn + β (R+1/2 )n︱-︱1+ β(R+1/2 )n︱) min|z|=1 |P(z)︱}.In this paper we generalize the above inequality for polynomials having no zeros in |z|相似文献   

Convergence of Gaussian Quadrature Formulas for Power Orthogonal Polynomials

In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I =（a,b）,a function G ∈ S（w）:= { f:∫_I | f（x）| w（x）d x < ∞} satisfying the conditions G ^2j（x） ≥ 0,x ∈（a,b）,j = 0,1,...,and growing as fast as possible as x → a + and x → b,plays an important role.But to find such a function G is often difficult and complicated.This implies that to prove convergence of Gaussian quadrature formulas,it is enough to find a function G ∈ S（w） with G ≥ 0 satisfying sup n ∑λ0knG（xkn） k=1 n<∞ instead,where the xkn ’s are the zeros of the n th power orthogonal polynomial with respect to the weight w and λ0kn ’s are the corresponding Cotes numbers.Furthermore,some results of the convergence for Gaussian quadrature formulas involving the above condition are given.     

REGULARITY AND EXPLICIT REPRESENTATION OF (0,1,…,m - 2,m) INTERPOLATION ON THE ZEROS OF (1-x)P_(n-1)~(α,β)(x)

A necessary and sufficient condition of regularity of (0,1,…,m - 2,m) interpolation on the zeros of (1-x)P_(n-1)~(α,β)(x) (α> -1,β≥- 1) in a manageable form is established, where P_(n-1)~(α,β)(x) stands for the (n-1)th Jacobi polynomial. Meanwhile, the explicit representation of the fundamental polynomials when they exist, is given.     

ON THE LORENTZ CONJECTURES UNDER THE L_1-NORM

Let f (x) ∈ C [-1, 1], p_n~* (x) be the best approximation polynomial of degree n tof (x). G. Iorentz conjectured that if for all n, p_(2n)~* (x) = p_(2n+1)~* (x), then f is even; and ifp_(2n+1)~* (x) = p_(2n+2)~* (x), p_o~* (z) = 0, then f is odd. In this paper, it is proved that, under the L_1-norm, the Lorentz conjecture is validconditionally, i. e. if (i) (1-x~2) f (x) can be extended to an absolutely convergentTehebyshev sories; (ii) for every n, f (x) - p_(2n+1)~* (x) has exactly 2n + 2 zeros (or, in thearcond situation, f (x) - p_(2n+2)~* (x) has exaetly 2n+3 zeros), then Lorentz conjecture isvalid.     

Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight

In this paper,the authors consider the asymptotic behavior of the monic polynomials orthogonal with respect to the weight function w(x) = |x|~(2α)e~(-(x~4+tx~2)),x ∈ R,where α is a constant larger than -1/2 and t is any real number. They consider this problem in three separate cases:(i) c -2,(ii) c =-2,and(iii) c -2,where c := t N~(-1/2) is a constant,N = n + α and n is the degree of the polynomial. In the first two cases,the support of the associated equilibrium measure μ_t is a single interval,whereas in the third case the support of μ_t consists of two intervals. In each case,globally uniform asymptotic expansions are obtained in several regions. These regions together cover the whole complex plane. The approach is based on a modified version of the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou(1993).     

Approximation to Continuous Functions by a Kind of Interpolation Polynomials

In this paper, an interpolation polynomial operator Fn (f; l, x ) is constructed based on the zeros of a kind of Jacobi polynomials as the interpolation nodes. For any continuous function. f(x)∈ C^b[-1,1] (0≤b≤1) Fn(f; l,x) converges to f(x) uniformly, where l is an odd number.     

THE CONVERGENCE ORDER OF THE INTERPOLATION PROCESS BY LAGRANGE POLYNOMIALS

Let f(x) be an arbitrary continuous function on [-1, 1] and letus denote T_n(x)=cos nθ, x=cos θ,T_n(x) is to be known as the first kind of Chebyshev polynomial ofdegree n. The zeros. of T_n(x) are     

The rate of the normal approximation for jackknifingU-statistics

Let U_n be a U-statistic with symmetric kernel h(x,y) such that Eh(X_1,X_2)=θ and Var E[h(X_1,X_2)-θ|X_j]>0.Let f(x) be a function defined on R and f″ be bounded.If f(θ) is the parameterof interest,a natural estimator is f(U_n).It is known that the distribution function of z_n=(n~(1/2){Jf(U_n)-f(θ)})/(S_n~*) converges to the standard normal distribution Φ(x) as n→∞,where Jf(U_n) isthe jackknife estimator of f(U_n),and S_n~(*2) is the jackknife estimator of the asymptotic variance ofn~(1/2) Jf(U_n).It is of theoretical value to study the rate of the normal approximation of the statistic.In this paper,assuming the existence of fourth moment of h(X_1,X_2),we show that(?)|P{z_n≤x}-Φ(x)|=O(n~(-1/2)log n).This improves the earlier results of Cheng(1981).     

关于α次的殆星映射推广的Roper-Suffridge算子的一个注记

Suppose f is an almost starlike function of order α on the unit disk D. In this paper, we will prove that Φn, β2, γ2, …, βn, γn (f)(z) = (f(z1), (f(z1)/z1)β2(f'(z1))γ2z2,…,(f(z1)/z1)βn(f'(z11))γnzn)' preserves almost starlikeness of order α on Ωn,p1,p2,…,pn = {z =(z1,z2,…,zn)' ∈ Cn n∑j=1 |zj|pj ＜ 1}, where 0 ＜ p11 ≤ 2, pj ≥ 1, j = 2,…,n, are real numbers.