A note on optimal Hermite interpolation in Sobolev spaces

This paper investigates the optimal Hermite interpolation of Sobolev spaces $W_{\infty }^{n}[a,b]$ W ∞ n [ a , b ] , $n\in \mathbb{N}$ n ∈ N in space $L_{\infty }[a,b]$ L ∞ [ a , b ] and weighted spaces $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p< \infty $ 1 ≤ p < ∞ with ω a continuous-integrable weight function in $(a,b)$ ( a , b ) when the amount of Hermite data is n. We proved that the Lagrange interpolation algorithms based on the zeros of polynomial of degree n with the leading coefficient 1 of the least deviation from zero in $L_{\infty }$ L ∞ (or $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p<\infty $ 1 ≤ p < ∞ ) are optimal for $W_{\infty }^{n}[a,b]$ W ∞ n [ a , b ] in $L_{\infty }[a,b]$ L ∞ [ a , b ] (or $L_{p,\omega }[a,b]$ L p , ω [ a , b ] , $1\le p<\infty $ 1 ≤ p < ∞ ). We also give the optimal Hermite interpolation algorithms when we assume the endpoints are included in the interpolation systems.

The Improved Interpolating Dimension Splitting Element-Free Galerkin Method for 3D Potential Problems

This paper proposes the improved interpolating dimension splitting element-free Galerkin (IIDSEFG) method based on the nonsingular weight function for three-dimensional (3D) potential problems. The core of the IIDSEFG method is to transform the 3D problem domain into a series of two-dimensional (2D) problem subdomains along the splitting direction. For the 2D problems on these 2D subdomains, the shape function is constructed by the improved interpolating moving least-squares (IIMLS) method based on the nonsingular weight function, and the finite difference method (FDM) is used to couple the discretized equations in the direction of splitting. Finally, the calculation formula of the IIDSEFG method for a 3D potential problem is derived. Compared with the improved element-free Galerkin (IEFG) method, the advantages of the IIDSEFG method are that the shape function has few undetermined coefficients and the essential boundary conditions can be executed directly. The results of the selected numerical examples are compared by the IIDSEFG method, IEFG method and analytical solution. These numerical examples illustrate that the IIDSEFG method is effective to solve 3D potential problems. The computational accuracy and efficiency of the IIDSEFG method are better than the IEFG method.

Equality of Ultradifferentiable Classes by Means of Indices of Mixed O-regular Variation

We characterize the equality between ultradifferentiable function classes defined in terms of abstractly given weight matrices and in terms of the corresponding matrix of associated weight functions by using new growth indices. These indices, defined by means of weight sequences and (associated) weight functions, are extending the notion of O-regular variation to a mixed setting. Hence we are extending the known comparison results concerning classes defined in terms of a single weight sequence and of a single weight function and give also these statements an interpretation expressed in O-regular variation.

Cosheaf representations of relations and Dowker complexes

The Dowker complex is an abstract simplicial complex that is constructed from a binary relation in a straightforward way. Although there are two ways to perform this construction—vertices for the complex are either the rows or the columns of the matrix representing the relation—the two constructions are homotopy equivalent. This article shows that the construction of a Dowker complex from a relation is a non-faithful covariant functor. Furthermore, we show that this functor can be made faithful by enriching the construction into a cosheaf on the Dowker complex. The cosheaf can be summarized by an integer weight function on the Dowker complex that is a complete isomorphism invariant for the relation. The cosheaf representation of a relation actually embodies both Dowker complexes, and we construct a duality functor that exchanges the two complexes.

ON THE PROBLEM OF RESTORING A FUNCTION IN THREE DIMENSIONAL SPACE BY THE FAMILY OF CONES

In this paper, we consider the problem of recovering a function in three-dimensional space from a family of cones with a weight function of a special form. Exact solutions of the problem are obtained for the given weight functions. A class of parameters for the problem that has no solution is constructed.

Orthogonal Polynomials on Planar Cubic Curves

Orthogonal polynomials in two variables on cubic curves are considered. For an integral with respect to an appropriate weight function defined on a cubic curve, an explicit basis of orthogonal polynomials is constructed in terms of two families of orthogonal polynomials in one variable. We show that these orthogonal polynomials can be used to approximate functions with cubic and square root singularities, and demonstrate their usage for solving differential equations with singular solutions.

Global structure for a fourth-order boundary value problem with sign-changing weight

We study the fourth-order boundary value problem with a sign-changing weight function: u ⁗ = λ m ( t ) u + f 1 ( t , u , u ′ , u ″ , u ‴ , λ ) + f 2 ( t , u , u ′ , u ″ , u ‴ , λ ) , t ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , $$\left\{\begin{array}{ll} u''''=\lambda m(t)u+f_1(t,u,u',u'',u''',\lambda)+f_2(t,u,u',u'',u''',\lambda),\qquad t\in(0,1),\\[1.3ex] u(0)=u(1)=u''(0)=u''(1)=0, \end{array}\right.$$ where λ ∈ ℝ is a parameter, f 1, f 2 ∈ C([0, 1] × ℝ5, ℝ), f 1 is not differentiable at the origin and infinity. Under some suitable conditions on nonlinear terms, we prove the existence of unbounded continua of positive and negative solutions of this problem which bifurcating from intervals of the line of trivial solutions or from infinity, respectively.