ON LAGRANGE INTERPOLATION TO |x|α(1 ＜α＜ 2) WITH EQUALLY SPACED NODES

S.M. Lozinskii proved the exact convergence rate at the zero of Lagrange interpolation polynomials to |x| based on equidistant nodes in [-1, 1]. In 2000, M. Rever generalized S.M. Lozinskii's result to |x|α(0 ≤α≤ 1). In this paper we will present the exact rate of convergence at the point zero for the interpolants of |x|α(1 ＜α＜ 2)..     

THE EXACT CONVERGENCE RATE AT ZERO OF LAGRANGE INTERPOLATION POLYNOMIAL TO ｜x｜^α

In this paper we present a generalized quantitative version of a result the exact convergence rate at zero of Lagrange interpolation polynomial to spaced nodes in [-1,1] due to M.Revers concerning f（x） = ｜x｜α with on equally     

THE DIVERGENCE OF LAGRANGE INTERPOLATION FOR |x|~a

This paper shows that the sequence of Lagrange interpolation polynomials corresponding to the function f(x) =|x|~a(1相似文献   

SOME CONVERGENCE PROBLEMS REGARDING THE FRACTIONAL SCHR?DINGER PROPAGATOR ON NONCOMPACT MANIFOLDS

Let L be the Laplace-Beltrami operator.On an n-dimensional(n≥ 2),complete,noncompact Riemannian manifold M,we prove that if 0 <α <1,s> α/2 and f ∈ H^s(M),then the fractional Schr?dinger propagator e⁽it|L|^α/2)(f)(x)→f(x) a.e.as t→0.In addition,for when M is a Lie group,the rate of the convergence is also studied.These results are a non-trivial extension of results on Euclidean spaces and compact manifolds.     

CONVERGENCE RATE OF SOLUTIONS TO STRONG CONTACT DISCONTINUITY FOR THE ONE-DIMENSIONAL COMPRESSIBLE RADIATION HYDRODYNAMICS MODEL

This paper is concerned with a singular limit for the one-dimensional compressible radiation hydrodynamics model. The singular limit we consider corresponds to the physical problem of letting the Bouguer number infinite while keeping the Boltzmann number constant. In the case when the corresponding Euler system admits a contact discontinuity wave, Wang and Xie(2011) [12] recently verified this singular limit and proved that the solution of the compressible radiation hydrodynamics model converges to the strong contact discontinuity wave in the L∞-norm away from the discontinuity line at a rate of ε14, as the reciprocal of the Bouguer number tends to zero. In this paper, Wang and Xie's convergence rate is improved to ε~(7/8) by introducing a new a priori assumption and some refined energy estimates. Moreover, it is shown that the radiation flux q tends to zero in the L∞-norm away from the discontinuity line, at a convergence rate as the reciprocal of the Bouguer number tends to zero.     

光滑函数类中一族无高阶导数的大范围收敛迭代法

The problem whether the iteration formula with the global convergence which does notneed to compute the second order derivative of the function can be found, raised in [7], issolved for f(x)∈C~1(R~1) in the present paper by using the methods of prior estimates andintroducing a parametric function. The main results are as follows: 1. For f(x)∈C~1(R~1), the families of iteration formulas of the global convergence,without derivatives of higher order, are suggested in the following formx_(n 1)=x_n±|f(x_n)|/|f'(x_n)| α(x_n)|f(x_n)|,(1)x_(n 1)=x_n-α|f(x_n)|/(α-1)f'(x_n)sgnf(x_0)±(f'2(x_n)αp(x_n)|f(x_n)|),(2)x_(n 1)=x_n±|f(x_n)f'(x_n)|/f'2(x_n) 1/2p(x_n)|f(x_n)|,(3)Where the real parameter a∈(0, 2] and the real parametric functions α(x)=α(f(x),f'(x)) (>0) and p(x)= p(f(x), f,(x)) (>0) with certain arbitrariness are continuous orpiecewise continuous. 2. The convergence order of the iteration sequence {x_n} generated by (1), (2) or (3)is 2 for a simple real zero of f(x), and is 1 for a multiple zero.     

THE ZERO LIMIT OF THERMAL DIFFUSIVITY FOR THE 2D DENSITY-DEPENDENT BOUSSINESQ EQUATIONS

This paper is concerned with the asymptotic behavior of solutions to the initial boundary problem of the two-dimensional density-dependent Boussinesq equations.It is shown that the solutions of the Boussinesq equations converge to those of zero thermal diffusivity Boussinesq equations as the thermal diffusivity tends to zero,and the convergence rate is established.In addition,we prove that the boundary-layer thickness is of the valueδ(k)=k^α with any α∈(0,1/4) for a small diffusivity c...     

一类离散分布参数的经验Bayes估计的收敛速度

In this paper we consider a family of discrete distributions f_θ(x)dμ(x), and suppose that the Bayes estimate of φ(θ) with respect to the priori distribution H∈H has a form dH(x) =(?)a_k(x)f(x+k)/f(x). where f(x)=∫f_θdH(θ) . we construct asequence of empirical Bayes estimates and establish its rate of convergence, and prove that under suitable conditions this rate of convergence can arbitrarily close to 1. we also give a counter-example to the main Theorem 2.1 of [5], and then declare that the "Theorem" does not hold.     

BOUND STATES FOR A CLASS OF QUASILINEAR SCALAR FIELD EQUATIONS WITH POTENTIALS VANISHING AT INFINITYDedicated to Professor Constantine M. Dafermos on the occasion of his 70th birthday

We study the existence and non-existence of bound states(i.e.,solutions in W1,p(RN)) for a class of quasilinear scalar field equations of the form-△pu+V(x)|u|p-2 u=a(x)|u|q-2 u,x∈RN,1相似文献   

CONVERGENCE OF DERIVATIVES OF GENERALIZED BERNSTEIN OPERATORS

In the present paper, we obtain estimations of convergence rate derivatives of the q-Bernstein polynomials Bn (f, qn ; x) approximating to f ′ (x) asn →∞, which is a generalization of that relating the classical case qn = 1. On the other hand, we study the convergence properties of derivatives of the limit q-Bernstein operators B ∞ (f, q; x) as q → 1- .     

ON THE ORDER OF APPROXIMATION FOR THE RATIONAL INTERPOLATION TO |x|

The order of approximation for Newman-type rational interpolation to |x| is studied in this paper. For general set of nodes, the extremum of approximation error and the order of the best uniform approximation are estimated. The result illustrates the general quality of approximation in a different way. For thespecial case where the interpolation nodes are xi= (i/n)(i= 1,2,…,n;r>0) , it is proved that the exact order of approximation is O(1/n),O(1/nlogn) and O(1/n), respectively, corresponding to O1.     

On approximation properties of non-convolution type nonlinear integral operators

In the present paper we state some approximation theorems concerning pointwise convergence and its rate for a class of non-convolution type nonlinear integral operators of the form:Tλ (f;x) = B A Kλ (t,x, f (t))dt , x ∈< a,b >, λ∈Λ. In particular, we obtain the pointwise convergence and its rate at some characteristic points x0 of f as (x,λ ) → (x0,λ0) in L1 < A,B >, where < a,b > and < A,B > are is an arbitrary intervals in R, Λ is a non-empty set of indices with a topology and λ0 an accumulation point of Λ in this topology. The results of the present paper generalize several ones obtained previously in the papers [19]-[23].     

Optimal time-decay rates of the Boltzmann equation

We consider the optimal time-convergence rates of the global solution to the Cauchy problem for the Boltzmann equation in R3.We show that the global solution tends to the global Maxwellian at the optimal time-decay rate(1+t)-3/4,where the macroscopic density,momentum and energy decay at the optimal rate(1+t)-3/4 and the microscopic part decays at the optimal rate(1+t)-5/4.We also show that the solution tends to the Maxwellian at the optimal time-decay rate(1+t).5/4 in the case of the macroscopic part of the initial data is zero,where the macroscopic density,momentum and energy decay at the optimal rate(1+t)-5/4 and the microscopic part decays at the optimal rate(1+t)-7/4.These convergence rates are shown to be optimal for the Boltzmann equation.     

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.     

Existence and Concentration of Bound States of a Class of Nonlinear Schrdinger Equations in R~2 with Potential Tending to Zero at Infinity

In this paper, we establish the existence and concentration of solutions of a class of nonlinear Schrdinger equation -ε2 Δuε + V(x)uε = K(x)|uε|p-2 uεeα0 |uε|γ,uε0, uε∈H 1(R2),where 2 p ∞, α0 0, 0 γ 2. When the potential function V (x) decays at infinity like (1 + |x|)-α with 0 α≤ 2 and K(x) 0 are permitted to be unbounded under some necessary restrictions, we will show that a positive H1 (R2 )-solution uε exists if it is assumed that the corresponding ground energy function G(ξ) of nonlinear Schrdinger equation-Δu + V (ξ)u = K(ξ)|u| p-2 ue α0 |u|γ has local minimum points. Furthermore, the concentration property of uε is also established as ε tends to zero.     

条件密度双重核估计误差分布的逼近速度

In this paper, the normal approximation rate and the random weighting approximation rate of error distribution of the kernel estimator of conditional density function f(y!|x) are studied. The results may be used to construct the confidence interval of f(y|x).     

BMO ESTIMATES FOR MULTILINEAR FRACTIONAL INTEGRALS

In this paper,the authors prove that the multilinear fractional integral operator T A 1,A 2 ,α and the relevant maximal operator M A 1,A 2 ,α with rough kernel are both bounded from L p (1 p ∞) to L q and from L p to L n/(n α),∞ with power weight,respectively,where T A 1,A 2 ,α (f)(x)=R n R m 1 (A 1 ;x,y)R m 2 (A 2 ;x,y) | x y | n α +m 1 +m 2 2 (x y) f (y)dy and M A 1,A 2 ,α (f)(x)=sup r0 1 r n α +m 1 +m 2 2 | x y | r 2 ∏ i=1 R m i (A i ;x,y)(x y) f (y) | dy,and 0 α n, ∈ L s (S n 1) (s ≥ 1) is a homogeneous function of degree zero in R n,A i is a function defined on R n and R m i (A i ;x,y) denotes the m i t h remainder of Taylor series of A i at x about y.More precisely,R m i (A i ;x,y)=A i (x) ∑ | γ | m i 1 γ ! D γ A i (y)(x y) r,where D γ (A i) ∈ BMO(R n) for | γ |=m i 1(m i 1),i=1,2.     

ON THE RATE OF POINTWISE CONVERGENCE OF THE DURRMEYER-TYPE OPERATORS

For bounded or some locally bounded functions f measurable on an interval I there is estimated the rate of convergence of the Durrmeyer-type operators Lnf at those points x∈IntI at which the one-sided limits f(x± 0) exist. In the main theorems the Chanturiya's modulus of variation is used.     

ON THE RIGOROUS MATHEMATICAL DERIVATION FOR THE VISCOUS PRIMITIVE EQUATIONS WITH DENSITY STRATIFICATION

In this paper, we rigorously derive the governing equations describing the motion of a stable stratified fluid, from the mathematical point of view. In particular, we prove that the scaled Boussinesq equations strongly converge to the viscous primitive equations with density stratification as the aspect ratio goes to zero, and the rate of convergence is of the same order as the aspect ratio. Moreover, in order to obtain this convergence result, we also establish the global well-posedness of stro...     

THE DIVERGENCE OF LAGRANGE INTERPOLATION FOR ｜x｜^α（2〈α〈4）AT EQUIDISTANT NODES

It is a classical result of Bernstein that the sequence of Lagrange interpolation polumomials to ｜x｜ at equally spaced nodes in [-1, 1] diverges everywhere, except at zero and the end-points. In the present paper, toe prove that the sequence of Lagrange interpolation polynomials corresponding to ｜x｜^α （2 〈 α 〈 4） on equidistant nodes in [-1, 1] diverges everywhere, except at zero and the end-points.