On the rigidity theorems for lagrangian translating solitons in pseudo-Euclidean space I

Let f be a smooth strictly convex solution of
defined on a domain Ω C R^n, where ai, bi and c are constants. Then the graph Mvf of △f is a spacelike translating soliton for mean curvature flow in pseudo-Euclidean space with the translating vector（al, a2, . ., an; bz, b2, , bn）. In this paper, we will use alCfine technique to show a Berustein Theorem： if the graph Mvf is complete, then f（x） must be a quadratic polynomial and Mvf is an ailine n-plane.     

Approximation of Generalized Bernstein Operators

This paper is devoted to study direct and converse approximation theorems of the generalized Bernstein operators Cn( f,sn,x) via so-called unified modulus ω2φλ( f,t), 0 ≤λ≤1. We obtain main results as follows ω2φλ( f,t) =O(tα)|Cn( f,sn,x)- f(x)| =O(n-12 δ1-λn(x))α,where δ2n(x) =max{φ2(x),1/n} and 0 α 2.     

The saturation class for linear combinations of Bernstein operators

In this paper we study the saturation class for the linear combinations of Bernstein operators. The characterization of the saturation class involving the modulus of smoothness is proved under certain assumption. Received: 1 September 2007     

Invariance of dimension of p-subspaces in bernstein algebras *

The purpose of this paper is to prove that some vector subspaces, called p-subspaces, obtained from the Peirce decomposition of a Bernstein algebra A relative to an idempotent have dimensions which are independent of the idempotent used to decompose A. In particular, for Bernstein-Jordan algebras, this fact is true for every such subspace and this implies that all p-subspaces of a Bernstein algebra, contained in V, for A = Ke + U + V, have invariant dimension. Finally we classify all p-subspaces of degree ≥ 3, contained in U, in a Bernstein algebra A, relative to the invariance (or not) of dimension.     

Degree of approximation associated with some elliptic operators and its applications

The Jackson-type estimates by using some elliptic operators will be achieved. These results will be used to characterize the regularity of some elliptic operators by means of the approximation degree and the saturation class of the multivariate Bernstein operators as well.     

单纯形上的q-Stancu多项式的最优逼近阶

构造了单纯形上的多元q-Stancu多项式,它是著名的Bernstein多项式和Stancu多项式的推广.建立该类多项式逼近连续函数的上、下界估计,进而给出其对连续函数的最优逼近阶(饱和阶)及其特征刻画.此外,还研究了该类多项式逼近连续函数的饱和类.     

Bernstein Fractal Trigonometric Approximation

Fractal interpolation and approximation received a lot of attention in the last thirty years. The main aim of the current article is to study a fractal trigonometric approximants which converge to the given continuous function even if the magnitude of the scaling factors does not approach zero. In this paper, we first introduce a new class of fractal approximants, namely, Bernstein \(\alpha \)-fractal functions using the theory of fractal approximation and Bernstein polynomial. Using the proposed class of fractal approximants and imposing no condition on corresponding scaling factors, we establish that the set of Bernstein \(\alpha \)-fractal trigonometric functions is fundamental in the space of continuous periodic functions. Fractal version of Gauss formula of trigonometric interpolation is obtained by means of Bernstein trigonometric fractal polynomials. We study the Bernstein fractal Fourier series of a continuous periodic function \(f\) defined on \([-l,l]\). The Bernstein fractal Fourier series converges to \(f\) even if the magnitude of the scaling factors does not approach zero. Existence of the \(\mathcal{C}^{r}\)-Bernstein fractal functions is investigated, and Bernstein cubic spline fractal interpolation functions are proposed based on the theory of \(\mathcal{C}^{r}\)-Bernstein fractal functions.

     

Local saturation of conservative operators

We state a result about the local saturation of sequences of linear operators that preserve the sign of the k-th derivative of the functions. We apply it to the well known approximation operators of Bernstein, Szász–Mirakjan, Meyer–König and Zeller, and Bleimann, Butzer and Hahn.     

A note on a result of Zolotarev and Bernstein

In this note we obtain error bounds to x²ⁿ⁺¹–x²ⁿ on [–1, 1] by polynomials of degree at most (2n–1). The result proved here improves and extends some of the known results of Zolotarev and Bernstein. The proof presented here is different (and simple) from the one adopted by Zolotarev and Bernstein.     

Critical regularity for elliptic equations from Littlewood-Paley theory

Using simple facts from harmonic analysis, namely Bernstein inequality and Plansherel isometry, we prove that the pseudo differential equation improves the Sobolev regularity of solutions provided the potential V is integrable with the critical power .     

Complete Asymptotic Expansions for the Chlodovsky Polynomials

The purpose of this article is to study the local rate of convergence of the Chlodovsky operators (C_nf)(x). As the main results, we investigate their asymptotic behaviour and derive the complete asymptotic expansions of these operators. All the coefficients of n^?k (k = 1, 2,…) are calculated in terms of the Stirling numbers of first and second kind. We mention that analogous results for the Bernstein polynomials can be found in Lorentz [2 G. G. Lorentz ( 1953 ). Bernstein Polynomials . University of Toronto Press , Toronto . [Google Scholar]].     

Classification of bernstein algebras of type (3, n - 3)

A classification of Bernstein algebras in dimensions n ? 4 has been made by Holgate in [2], however that article contains no classification up to isomorphism, the problem is solved by Lyubich in [4] when K = R or C, and by Cortes [1] in the general case. Also Lyubich has given in [5] a classification of the regular nonexceptional Bernstein algebra of type (3,n?3) and a classification but not up to isomorphism of nonregular nonexceptional Bernstein algebras of type (3,n ? 3) when K = C. The aim of this paper it to characterize, up to isomorphism, Bernstein algebras of type(2, n ? 2) and nonexceptional of type(3, n ?3) over a infinite commutative field K whose characteristic is different from 2.     

Convexity and Bernstein polynomials onk-simploids

This paper is concerned with Bernstein polynomials onk-simploids by which we mean a cross product ofk lower dimensional simplices. Specifically, we show that if the Bernstein polynomials of a given functionf on ak-simploid form a decreasing sequence thenf +l, wherel is any corresponding tensor product of affine functions, achieves its maximum on the boundary of thek-simploid. This extends recent results in [1] for bivariate Bernstein polynomials on triangles. Unlike the approach used in [1] our approach is based on semigroup techniques and the maximum principle for second order elliptic operators. Furthermore, we derive analogous results for cube spline surfaces.This work was partially supported by NATO Grant No. DJ RG 639/84.     

The Genuine Bernstein-Durrmeyer Operator on a Simplex

In 1967 Durrmeyer introduced a modification of the Bernstein polynomials as a selfadjoint polynomial operator on L²[0,1] which proved to be an interesting and rich object of investigation. Incorporating Jacobi weights Berens and Xu obtained a more general class of operators, sharing all the advantages of Durrmeyer’s modification, and identified these operators as de la Vallée-Poussin means with respect to the associated Jacobi polynomial expansion. Nevertheless, all these modifications lack one important property of the Bernstein polynomials, namely the preservation of linear functions. To overcome this drawback a Bernstein-Durrmeyer operator with respect to a singular Jacobi weight will be introduced and investigated. For this purpose an orthogonal series expansion in terms generalized Jacobi polynomials and its de la Vallée-Poussin means will be considered. These Bernstein-Durrmeyer polynomials with respect to the singular weight combine all the nice properties of Bernstein-Durrmeyer polynomials with the preservation of linear functions, and are closely tied to classical Bernstein polynomials. Focusing not on the approximation behavior of the operators but on shape preserving properties, these operators we will prove them to converge monotonically decreasing, if and only if the underlying function is subharmonic with respect to the elliptic differential operator associated to the Bernstein as well as to these Bernstein-Durrmeyer polynomials. In addition to various generalizations of convexity, subharmonicity is one further shape property being preserved by these Bernstein-Durrmeyer polynomials. Finally, pointwise and global saturation results will be derived in a very elementary way.     

On approximation by some Bernstein–Kantorovich exponential-type polynomials

Periodica Mathematica Hungarica - Since the introduction of Bernstein operators, many authors defined and/or studied Bernstein type operators and their generalizations, among them are Morigi and...     

The combinatorics of Bernstein functions

A construction of Bernstein associates to each cocharacter of a split -adic group an element in the center of the Iwahori-Hecke algebra, which we refer to as a Bernstein function. A recent conjecture of Kottwitz predicts that Bernstein functions play an important role in the theory of bad reduction of a certain class of Shimura varieties (parahoric type). It is therefore of interest to calculate the Bernstein functions explicitly in as many cases as possible, with a view towards testing Kottwitz' conjecture. In this paper we prove a characterization of the Bernstein function associated to a minuscule cocharacter (the case of interest for Shimura varieties). This is used to write down the Bernstein functions explicitly for some minuscule cocharacters of ; one example can be used to verify Kottwitz' conjecture for a special class of Shimura varieties (the ``Drinfeld case'). In addition, we prove some general facts concerning the support of Bernstein functions, and concerning an important set called the ``-admissible' set. These facts are compatible with a conjecture of Kottwitz and Rapoport on the shape of the special fiber of a Shimura variety with parahoric type bad reduction.

     

A recursion formula for k-Schur functions

The Bernstein operators allow one to build recursively the Schur functions. We present a recursion formula for k-Schur functions at t=1 based on combinatorial operators that generalize the Bernstein operators. The recursion leads immediately to a combinatorial interpretation for the expansion coefficients of k-Schur functions at t=1 in terms of homogeneous symmetric functions.