A shape-preserving quasi-interpolation operator satisfying quadratic polynomial reproduction property to scattered data

In this paper, we construct a univariate quasi-interpolation operator to non-uniformly distributed data by cubic multiquadric functions. This operator is practical, as it does not require derivatives of the being approximated function at endpoints. Furthermore, it possesses univariate quadratic polynomial reproduction property, strict convexity-preserving and shape-preserving of order 3 properties, and a higher convergence rate. Finally, some numerical experiments are shown to compare the approximation capacity of our quasi-interpolation operator with that of Wu and Schaback’s quasi-interpolation scheme.     

Some classes of functions of a linear closed operator

A linear closed densely defined operator and some domain Ω lying in the regular set of the operator and containing the negative real semiaxis of the real line are specified in a Banach space. We assume that power estimates for the norm of the resolvent operator are known at zero and infinity. We use the Cauchy integral formula to introduce operator functions generated by scalar functions that are analytic in a certain domain not containing the origin and containing the complement of Ω and satisfy power estimates for their absolute values at zero and infinity. We study some properties of operator functions, which were studied by the authors earlier for the case of an operator whose inverse is bounded; in particular, we study the multiplicative property.     

Singular value expansion for the Green function of Helmholtz operator

We consider the integral operator defined on a circular disk, and with kernel the Green function of the Helmholtz operator. We present an analytic framework for the explicit computation of the singular system of this kernel. In particular, the main formulas of this framework are given by a characteristic equation for the singular values and explicit expressions for the corresponding singular functions. We provide also a property of the singular values, that gives an important information for the numerical evaluation of the singular system. Finally, we present a simple numerical experiment, where the singular system computed by a simple implementation of these analytic formulas is compared with the singular system obtained by a discretization of the Green function of the Helmholtz operator.     

Hamilton算子的一类n次数值域的对称性

以Hamilton算子的数值域为基础,研究了一类算子的二次数值域关于实轴,虚轴的对称性.此外从α-J-自伴算子的n次数值域关于过原点直线对称出发,得到了有界Hamilton算子的一类n次数值域关于虚轴的对称性.     

无穷维Hamilton算子的二次数值域

研究了一类无界无穷维Hamilton算子的二次数值域的性质,进而,应用二次数值域来刻画了无穷维Hamilton算子谱的分布范围,并给出了二次数值域的闭包包含谱集的结论.     

Observability from measurable sets for a parabolic equation involving the Grushin operator and applications

In this work, we utilize the existing Carleman estimates and propagation estimates of smallness from measurable sets for real analytic functions, together with the telescoping series method, to establish an observability inequality from measurable subsets in time‐space variable for the parabolic equation with Grushin operator in some multidimension domains. We can apply this observability inequality to show the bang–bang property for both time optimal and norm optimal control problems for this kind of singular parabolic equation. Copyright © 2016 John Wiley & Sons, Ltd.     

A new concept for block operator matrices:the quadratic numerical range

In this paper a new concept for 2×2-block operator matrices – the quadratic numerical range – is studied. The main results are a spectral inclusion theorem, an estimate of the resolvent in terms of the quadratic numerical range, factorization theorems for the Schur complements, and a theorem about angular operator representations of spectral invariant subspaces which implies e.g. the existence of solutions of the corresponding Riccati equations and a block diagonalization. All results are new in the operator as well as in the matrix case.     

Properties of eigenvalues and spectral singularities for impulsive quadratic pencil of difference operators

In this paper, we investigate the spectral analysis of impulsive quadratic pencil of difference operators. We first present a boundary value problem consisting one interior impulsive point on the whole axis corresponding to the above mentioned operator. After introducing the solutions of impulsive quadratic pencil of difference equation, we obtain the asymptotic equation of the function related to the Wronskian of these solutions to be helpful for further works, then we determine resolvent operator and continuous spectrum. Finally, we provide sufficient conditions guarenteeing finiteness of eigenvalues and spectral singularities by means of uniqueness theorems of analytic functions. The main aim of this paper is demonstrating the impulsive quadratic pencil of difference operator is of finite number of eigenvalues and spectral singularities with finite multiplicities which is an uninvestigated problem proposed in the literature.     

Unique Continuation on Quadratic Curves for Harmonic Functions*

The unique continuation on quadratic curves for harmonic functions is discussed in this paper. By using complex extension method, the conditional stability of unique continuation along quadratic curves for harmonic functions is illustrated. The numerical algorithm is provided based on collocation method and Tikhonov regularization.The stability estimates on parabolic and hyperbolic curves for harmonic functions are demonstrated by numerical examples respectively.     

Remarks on the numerical range with respect to a family of projections

In this note we report on the recently introduced concept of the numerical range of a bounded linear operator on a Hilbert space with respect to a family of projections. The importance of this new concept lies in the fact that it unifies and generalizes well-established versions of the numerical range such as the classical numerical range introduced by Toeplitz and Hausdorff, the quadratic numerical range as well as the block numerical range. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Geometric and approximation properties of a complex Post–Widder operator in the unit disk

The purpose of this work is to obtain Jackson-type estimates in approximation by a complex Post–Widder operator in the unit disk. In addition, these operators preserve some sufficient conditions for starlikeness and univalence of analytic functions.     

Coefficient estimates for a certain family of analytic functions involving a q-derivative operator

The Ramanujan Journal - In this paper, we study the coefficient estimates for a class of analytic functions defined by using the q-Ruscheweyh derivative operator. In particular, we investigate the...     

Spectral inclusion for unbounded block operator matrices

In this paper we establish a new analytic enclosure for the spectrum of unbounded linear operators A admitting a block operator matrix representation. For diagonally dominant and off-diagonally dominant block operator matrices, we show that the recently introduced quadratic numerical range W²(A) contains the eigenvalues of A and that the approximate point spectrum of A is contained in the closure of W²(A). This provides a new method to enclose the spectrum of unbounded block operator matrices by means of the non-convex set W²(A). Several examples illustrate that this spectral inclusion may be considerably tighter than the one by the usual numerical range or by perturbation theorems, both in the non-self-adjoint case and in the self-adjoint case. Applications to Dirac operators and to two-channel Hamiltonians are given.     

INTERPOLATION INEQUALITIES AND DECAY ESTIMATES FOR DERIVATIVES

Under the only assumption of the cone property for a given domain Rn, it is proved that inter polation inequalities for intermediate derivatives of functions in the Sobolev spaces Wm,p or even in some weighted Sobolve spaces still hold. That is, the usual additional restrictions that is bounded or has the uniform cone property are both removed. The main tools used are polynomial inequalities, by which it is also ob- tained pointwise version inter polation inequalities for smooth and analytic functions. Such pointwise version in - equalities give explicit decay estimates for derivatives at infinity in unbounded domains which have the cone property. As an application of the decay ertimates, a previous result on radial basis function approximation of smooth functions is extended to the derivative-simultaneous approximation.     

Elliptic boundary value problems for Bessel operators,with applications to anti-de Sitter spacetimes

This paper considers boundary value problems for a class of singular elliptic operators that appear naturally in the study of asymptotically anti-de Sitter (aAdS) spacetimes. These problems involve a singular Bessel operator acting in the normal direction. After formulating a Lopatinski?? condition, elliptic estimates are established for functions supported near the boundary. The Fredholm property follows from additional hypotheses in the interior. This paper provides a rigorous framework for mode analysis on aAdS spacetimes for a wide range of boundary conditions considered in the physics literature. Completeness of eigenfunctions for some Bessel operator pencils is shown.     

与条形区域有关的双向单叶解析函数子类的系数估计

本文引入了条形区域内由Salagean算子刻画的解析函数新子类, 讨论了该函数子类的系数估计、Fekete-Szego不等式以及与该函数子类有关的双向单叶函数的系数估计, 所得结果推广了前人的一些工作.     

Continuous piecewise linear finite elements for the Kirchhoff–Love plate equation

A family of continuous piecewise linear finite elements for thin plate problems is presented. We use standard linear interpolation of the deflection field to reconstruct a discontinuous piecewise quadratic deflection field. This allows us to use discontinuous Galerkin methods for the Kirchhoff–Love plate equation. Three example reconstructions of quadratic functions from linear interpolation triangles are presented: a reconstruction using Morley basis functions, a fully quadratic reconstruction, and a more general least squares approach to a fully quadratic reconstruction. The Morley reconstruction is shown to be equivalent to the basic plate triangle (BPT). Given a condition on the reconstruction operator, a priori error estimates are proved in energy norm and L ² norm. Numerical results indicate that the Morley reconstruction/BPT does not converge on unstructured meshes while the fully quadratic reconstruction show optimal convergence.     

无界分块算子矩阵的谱分布

本文首先给出次对角元有界的2×2阶无界算子矩阵的Gershgorin定理,然后利用主对角元算子的谱和数值域刻画整个算子矩阵的谱分布.特别地,当次对角元算子互为共轭(反共轭)算子时,结合二次数值域和Gershgorin定理对谱分布给出更精细的描述.     

On the a posteriori estimates for inverse operators of linear parabolic equations with applications to the numerical enclosure of solutions for nonlinear problems

We consider the guaranteed a posteriori estimates for the inverse parabolic operators with homogeneous initial-boundary conditions. Our estimation technique uses a full-discrete numerical scheme, which is based on the Galerkin method with an interpolation in time by using the fundamental solution for semidiscretization in space. In our technique, the constructive a priori error estimates for a full discretization of solutions for the heat equation play an essential role. Combining these estimates with an argument for the discretized inverse operator and a contraction property of the Newton-type formulation, we derive an a posteriori estimate of the norm for the infinite-dimensional operator. In numerical examples, we show that the proposed method should be more efficient than the existing method. Moreover, as an application, we give some prototype results for numerical verification of solutions of nonlinear parabolic problems, which confirm the actual usefulness of our technique.     

Efficient computable error bounds for discontinuous Galerkin approximations of elliptic problems

We present guaranteed and computable both sided error bounds for the discontinuous Galerkin (DG) approximations of elliptic problems. These estimates are derived in the full DG-norm on purely functional grounds by the analysis of the respective differential problem, and thus, are applicable to any qualified DG approximation. Based on the triangle inequality, the underlying approach has the following steps for a given DG approximation: (1) computing a conforming approximation in the energy space using the Oswald interpolation operator, and (2) application of the existing functional a posteriori error estimates to the conforming approximation. Various numerical examples with varying difficulty in computing the error bounds, from simple problems of polynomial-type analytic solution to problems with analytic solution having sharp peaks, or problems with jumps in the coefficients of the partial differential equation operator, are presented which confirm the efficiency and the robustness of the estimates.