Recurrence Relations for the Coefficients in Series Expansions with Respect to Semi-Classical Orthogonal Polynomials

Let {P _k } be a sequence of the semi-classical orthogonal polynomials. Given a function f satisfying a linear second-order differential equation with polynomial coefficients, we describe an algorithm to construct a recurrence relation satisfied by the coefficients a _k [f] in f= _k a _k [f]P _k . A systematic use of basic properties (including some nonstandard ones) of the polynomials {P _k } results in obtaining a recurrence of possibly low order. Recurrences for connection or linearization coefficients related to the first associated generalized Gegenbauer, Bessel-type and Laguerre-type polynomials are given explicitly.     

Product of functions in BMO and <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript> in non-homogeneous spaces

Under the assumption that the underlying measure is a non-negative Radon measure which only satisfies some growth condition and may not be doubling, we define the product of functions in the regular BMO and the atomic block H ¹ in the sense of distribution, and show that this product may be split into two parts, one in L ¹ and the other in some Hardy-Orlicz space.     

The hardy and bellman operators in spaces connected with H( \mathbb{T} ) and BMO( \mathbb{T} )

Assume that 1 ≤ p < ∞ and a function f ∈ L ^p [0, π] has the Fourier series $ \sum\limits_{n = 1}^\infty {a_n } Assume that 1 ≤ p < ∞ and a function f ∈ L ^p [0, π] has the Fourier series cos nx. According to one result of G.H. Hardy, the series cos nx is the Fourier series for a certain function (f) ∈ L ^p [0, π]. But if 1 < p ≤ ∞ and f ∈ L ^p [0, π], then the series cos nx is the Fourier series for a certain function (f) ∈ L ^p [0, π]. Similar assertions are true for sine series. This allows one to define the Hardy operator on L ^p (), 1 ≤ p < ∞, and to define the Bellman operator on L ^p (), 1 < p ≤ ∞. In this paper we prove that the Bellman operator boundedly acts in VMO(), and the Hardy operator also maps a certain subspace C() onto VMO(). We also prove the invariance of certain classes of functions with given majorants of modules of continuity or best approximations in the spaces H(), L(), VMO() with respect to the Hardy and Bellman operators. Original Russian Text ? S.S. Volosivets and B.I. Golubov, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 5, pp. 4–13.     

Estimation of elastic parameters in beams and certain plates:H 1 regularization

The estimation of elastic parameters in beams and certain types of plates is discussed using anH ¹-regularization technique that easily accommodates pointwise constraints. The optimal coefficient is shown to enjoy more regularity than that assumed in the formulation of the problem. This additional smoothness is useful for analyzing the limit behavior of finite-dimensional problems. Numerical results are presented.This work was supported in part by the Air Force Office of Scientific Research, Grant AFOSR-84-0271.     

强阻尼波动方程的H1-Galerkin混合有限元超收敛分析

研究了强阻尼波动方程的H¹-Galerkin混合有限元方法的超收敛性. 借助于协调线性三角形元已有的分析估计式, 直接利用插值算子代替原始变量 u 的 Ritz 投影和应力变量 p 的 Ritz-Volterra 投影,对半离散和全离散格式, 得到了u在 H¹(Ω) 模和 p 在 H(div;Ω) 模意义下比以往文献高一阶的超逼近和超收敛结果.     

Continued Fractions for Rogers–Szegö Polynomials

We evaluate different Hankel determinants of Rogers–Szegö polynomials, and deduce from it continued fraction expansions for the generating function of RS polynomials. We also give an explicit expression of the orthogonal polynomials associated to moments equal to RS polynomials, and a decomposition of the Hankel form with RS polynomials as coefficients.     

Error estimates of H 1-Galerkin mixed finite element method for Schrödinger equation

An H^1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.     

关于L1空间多项式导数的估计

本义在加权Ｌ₁空间中对正导数多项式建立了精确的Ｂｅｒｎｓｔｃｉｎ型不等式．     

Approximation by Reciprocals of Polynomials with Positive Coefficients in L p Spaces

We prove that, if f(x) L ^p _[0,1], 1 < p < , f(x) 0, x [0,1], f 0, then there is a polynomial p(x) ⁺ _n such that f - 1/p _LP C(p)(f,n ^-1/2) _LP where ⁺ _n indicates the set of all polynomials of degree n with positive coeficients (see the definition (1) in the text).     

Connection coefficients between orthogonal polynomials and the canonical sequence: an approach based on symbolic computation

We deal with the problem of obtaining closed formulas for the connection coefficients between orthogonal polynomials and the canonical sequence. We use a recurrence relation fulfilled by these coefficients and symbolic computation with the Mathematica language. We treat the cases of Gegenbauer, Jacobi and a new semi-classical sequence.     

A Lower Bound for the L 1-norm of Fourier Series of Polynomial Type

Mathematical Notes -     

Superconvergence of H 1-Galerkin mixed finite element methods for parabolic problems

In this article, we study the semidiscrete H ¹-Galerkin mixed finite element method for parabolic problems over rectangular partitions. The well-known optimal order error estimate in the L ²-norm for the flux is of order 𝒪(h ^k+1) (SIAM J. Numer. Anal. 35 (2), (1998), pp. 712–727), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas element. We derive a superconvergence estimate of order 𝒪(h ^k+3) between the H ¹-Galerkin mixed finite element approximation and an appropriately defined local projection of the flux variable when k ≥ 1. A the new approximate solution for the flux with superconvergence of order 𝒪(h ^k+3) is realized via a postprocessing technique using local projection methods.     

Superconvergence of H 1-Galerkin Mixed Finite Element Methods for Second-Order Elliptic Equations

We derive superconvergence result for H ¹-Galerkin mixed finite element method for second-order elliptic equations over rectangular partitions. Compared to standard mixed finite element procedure, the method is not subject to the Ladyzhenskaya–Bab?ska–Brezzi (LBB) condition and the approximating finite element spaces are allowed to be of different polynomial degrees. Superconvergence estimate of order 𝒪(h ^k+3), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas elements, is established for the flux via a postprocessing technique.     

Weak-type Endpoint Estimates for Multilinear Singular Integral Operators

In this paper, the authors discuss a class of multilinear singular integrals and obtain that the operators are bounded from H^1(R^n) to weak L^1(R^n). Using this result, we can directly prove a main theorem in [5].     

Hardy spaces estimates for a class of multilinear homogeneous operators

It is proved that a class of multilinear singular integral operators with homogeneous kernels are bounded operators from the product spaces to the Hardy spacesH ^r , (ℝ ⁿ ) and the weak Hardy spaceH ^r,∞ (ℝ ⁿ . As an application of this result, the L ^p ,(ℝ ⁿ ) boundedness of a class of commutator for the singular integral with homogeneous kernels is obtained. Project supparted in part by the National Natural Science Foundation of Chind (Grant No. 19131080) of China and Doctoral Programme Foundation of Institution of Higher Education (Grant No. 98002703) of China.     

A posteriori error estimates for H 1-Galerkin mixed finite-element method for parabolic problems

The purpose of this article is to derive a posteriori error estimates for the H ¹-Galerkin mixed finite element method for parabolic problems. We study both semidiscrete and fully discrete a posteriori error analyses using standard energy argument. A fully discrete a posteriori error analysis based on the backward Euler method is analysed and upper bounds for the errors are derived. The estimators yield upper bounds for the errors which are global in space and time. Our analysis is based on residual approach and the estimators are free from edge residuals.     

Elliptic Obstacle Problems with Measurable Coefficients in Non-Smooth Domains

We establish a global weighted W ^{1, p} -regularity for solutions to variational inequalities and obstacle problems for divergence form elliptic systems with measurable coefficients in bounded non-smooth domains.