AN APPLICATION OF BERNSTEIN-DURRMEYER OPERATORS

In the present paper, we find that the Bernstein-Durrmeyer operators, besides their better applications in approximation theory and some other fields, are good tools in constructing translation network. With the help of the de la Vallée properties of the Bernstein-Durrmeyer operators a sequence of translation network operators is constructed and its degree of approximation is dealt.

APPLICATIONS OF THE BERNSTEIN-DURRMEYER OPERATORS IN ESTIMATING THE NORM OF MERCER KERNEL MATRICES

The paper is related to the norm estimate of Mercer kernel matrices. The lower and upper bound estimates of Rayleigh entropy numbers for some Mercer kernel matrices on [0, 1] × [0, 1] based on the Bernstein-Durrmeyer operator kernel are obtained, with which and the approximation property of the Bernstein-Durrmeyer operator the lower and upper bounds of the Rayleigh entropy number and the l2 -norm for general Mercer kernel matrices on [0, 1] x [0, 1] are provided.

On Weighted L^p-Approximation by Weighted Bernstein-Durrmeyer Operators

In the present paper, we establish direct and converse theorems for weight-ed Bernstein-Durrmeyer operators under weighted L^p-norm with Jacobi weight w（x）=x^a（1-x）β.All the results involved have no restriction a,β〈1-1/p,which indicates that the weighted Bemstein-Durrmeyer operators have some better approxi- mation properties than the usual Bernstein-Durrmeyer operators.

APPLICATIONS OF BERNSTEIN-DURRMEYER OPERATORS IN ESTIMATING THE COVERING NUMBER

The paper deals with estimates of the covering number for some Mercer kernel Hilbert space with Bernstein-Durrmeyer operators. We first give estimates of l2- norm of Mercer kernel matrices reproducing by the kernelsK（α,β）（x,y）：=∑∞k=0 Ck（α,β）（x）Qk（α,β）（y）,where Qk（α,β） （x） are the Jacobi polynomials of order k on （0, 1 ）, Ck（α,β） 〉 0 are real numbers, and from which give the lower and upper bounds of the covering number for some particular reproducing kernel Hilbert space reproduced by Kα,β （x, y）.

SOME APPROXIMATION THEOREMS FOR MODIFIED BERNSTEIN-DURRMEYER OPERATORS

The modified Bernstein-Durrmeyer operators discussed in this paper are given by M_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt, where We will show,for 0<α<1 and 1≤p≤∞ M,f-f_p=O(n^-a)ω_Φ~2(f,t)_p=O(t^(2a)), |M_n f-f(x)|≤M(x(1-x)/n+1/_n2)~a/2ω(f,t)=O(t^a), where otherwise.

Approximation Theorems of Bernstein-Durrmeyer Operators in L_M^（Ba） Spaces

With the weighted modulus of smoothness as a metric,we prove the direct and the inverse theorems of approximation by Bernstein-Durrmeyer operators in LBa M spaces. Especially an approximation equivalent theorem of the operators is also obtained.

APPROXIMATION BY A CLASS OF AIODIFIED BERNSTEIN-DURRMEYER OPERATORS

We modify Bernstein-Durrmeyer operators by means of digonal matarix which overeome a difficulty in extending a Berens-Lorentz result to the Bernstein-Durrmeyer operators for second order of smoothness. The direct and converse theorems for these operators in L_p are also presented by Ditzian-Totik, modulus of smoothness.

Stechkin-Marchaud Type Inequalities in Lp for Linear Combination of Bernstein-Durrmeyer Operators

In this paper, we use the equivalence relation between K-functional and modulus of smoothness, and give the Stechkin-Marchaud-type inequalities for linear combination of Bernstein-Durrmeyer operators . Moreover, we obtain the inverse result of approximation for linear combination of Bernstein-Durrmeyer operators with . Meanwhile we unify and extend some previous results.

L^p APPROXIMATION BY GENERAL BERNSTEIN-DURRMEYER OPERATOR DEFINED ON SIMPLEX

As a generalization of the Bernstein-Durrmeyer operatora defined on the simplex, a class of general Bernstein-Durrmeyer operators is introduced. With the weighted moduli of smoothness as a metric, we prove a strong direct theorem and an inverse theorem of weak type for these operators by using a decom-position way. From the theorems the characterization of Lp approximation behavior is derived.

Passively Q-Switched Yb-Doped Fiber Laser Operating at 1.06 μm with Two-Dimensional Silver Nanoplate as Saturable Absorber

A two-dimensional silver nanoplate is prepared with the seed-mediated growth method and is used for achieving pulse fiber laser operation. By controlling the dimension parameters of the silver nanoplate, the surface plasmon resonance absorption peak of the material is successfully adjusted to 1068 nm. Based on the silver nanoplate as a saturable absorber, a passively Q-switched Yb-doped fiber laser operating at 1062 nm is demonstrated. The maximum average output power of 3.49mW is obtained with a minimum pulse width of 1.84#s at a pulse repetition rate of 65.TkHz, and the corresponding pulse energy and peak power are 53.1 nJ and 28.8mW, respectively.

Eigenfunction expansion method of upper triangular operator matrixand application to two-dimensional elasticity problems based onstress formulation

This paper studies the eigenfunction expansion method to solve the two dimensional （2D） elasticity problems based on the stress formulation. The fundamental system of partial differential equations of the 2D problems is rewritten as an upper tri angular differential system based on the known results, and then the associated upper triangular operator matrix matrix is obtained. By further research, the two simpler com plete orthogonal systems of eigenfunctions in some space are obtained, which belong to the two block operators arising in the operator matrix. Then, a more simple and conve nient general solution to the 2D problem is given by the eigenfunction expansion method. Furthermore, the boundary conditions for the 2D problem, which can be solved by this method, are indicated. Finally, the validity of the obtained results is verified by a specific example.

Simultaneous Approximation by Modified Bernstein-Durrmeyer Operators *

The aim of the present paper is to prove direct and converse results for simultaneous approximation by modified Bernstein-Durrmeyer operators. A point-wise equivalence characterization of simultaneous approximation is obtained.

THE UPWIND OPERATOR SPLITTING FINITE DIFFERENCE METHOD FOR COMPRESSIBLE TWO-PHASE DISPLACEMENT PROBLEM AND ANALYSIS

For compressible two-phase displacement problem, a kind of upwind operator splitting finite difference schemes is put forward and make use of operator splitting, of calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates are adopted. Optimal order estimates in L 2 norm are derived to determine the error, in the approximate solution.

Universal single-mode lasing in fully chaotic two-dimensional microcavity lasers under continuous-wave operation with large pumping power [Invited]

For a fully chaotic two-dimensional(2D) microcavity laser, we present a theory that guarantees both the existence of a stable single-mode lasing state and the nonexistence of a stable multimode lasing state, under the assumptions that the cavity size is much larger than the wavelength and the external pumping power is sufficiently large. It is theoretically shown that these universal spectral characteristics arise from the synergistic effect of two different kinds of nonlinearities: deformation of the cavity shape and mode interaction due to a lasing medium. Our theory is based on the linear stability analysis of stationary states for the Maxwell–Bloch equations and accounts for single-mode lasing phenomena observed in real and numerical experiments of fully chaotic 2D microcavitylasers.

Learning rates of least-square regularized regression with polynomial kernels

This paper presents learning rates for the least-square regularized regression algorithms with polynomial kernels. The target is the error analysis for the regression problem in learning theory. A regularization scheme is given, which yields sharp learning rates. The rates depend on the dimension of polynomial space and polynomial reproducing kernel Hilbert space measured by covering numbers. Meanwhile, we also establish the direct approximation theorem by Bernstein-Durrmeyer operators in $ L_{\rho _X }^2 $ with Borel probability measure.

Finite Difference Method for Inhomogeneous Fractional Dirichlet Problem

We make the split of the integral fractional Laplacian as(−△)^(s)u=(−△)(−△)^(s−1)u,where s∈(0,1/2)∪(1/2,1).Based on this splitting,we respectively discretize the oneand two-dimensional integral fractional Laplacian with the inhomogeneous Dirichlet boundary condition and give the corresponding truncation errors with the help of the interpolation estimate.Moreover,the suitable corrections are proposed to guarantee the convergence in solving the inhomogeneous fractional Dirichlet problem and an O(h^(1+α)2s))convergence rate is obtained when the solution u∈C^(1,α)(Ω_(n)^(δ)),where n is the dimension of the space,∈(max(0,2s−1),1],δis a fixed positive constant,and h denotes mesh size.Finally,the performed numerical experiments confirm the theoretical results.