The Best m-Term One-Sided Approximation of Besov Classes by the Trigonometric Polynomials

In this paper, we continue studying the so called best m-term one-sided approximation and Greedy-liked one-sided ap- proximation by the trigonometric polynomials. The asymptotic estimations of the best m-terms one-sided approximation by the trigonometric polynomials on some classes of Besov spaces in the metricLp(Td(1≤p≤∞ are given.

On Converse Theorem of Best Approximation by Polynomials in Bergman Spaces H_q^p(p>0,q>1)

A Bernstein type theorem and a converse theorem of best approximation by polynomials in Bergman spaces Hq^p(p>0,q>1) are proved.Some proofs and results in [1] are in proved.

THE BERNSTEIN TYPE INEQUALITY AND SIMULTANEOUS APPROXIMATION BY INTERPOLATION POLYNOMIALS IN COMPLEX DOMAIN

In this paper we investigate simultaneous approximation for arbitrary system of nodes on smooth domain in complex plane. Some results which are better than those of known theorems are obtained.

FIXED POINT THEOREMS AND BEST APPROXIMATION IN CONVEX METRIC SPACES

Results regarding best approximation and best Simultaneous approximation on convex metric spaces are Obtained.Existence of fixed points for an ultimately nonexpansive semigroup of mappings is also shown.

The Jackson Inequality for the Best L^2-Approximation of Functions on [0,1] with the Weight x

Let L^2([0,1],x) be the space of the real valued,measurable,square summable functions on [0,1] with weight x,and let ■_n be the subspace of L^2([0,1],x) defined by a linear combination of J_0(μ_kx),where J_0 is the Bessel function of order 0 and {μ_k} is the strictly increasing sequence of all positive zeros of J_0.For f∈L^2([0,1],x),let E(f,■_n) be the error of the best L^2([0,1],x),i.e.,approximation of f by elements of ■_n.The shift operator off at point x∈[0,1] with step t∈[0,1] is defined by T(t)f(x)=(1/π)∫_0~πf((x^2+t^2-2xtcosθ)^(1/2))dθ. The differences (1- T(t))^(r/2)f =∑_(j=0)~∞(-1)~j(_j^(r/2))T^j(t)f of order r∈(0,∞) and the L^2([0,1],x)-modulus of continuityω_r(f,τ)= sup{||(I-T(t))^(r/2)f||:0≤t≤τ}of order r are defined in the standard way,where T^0(t)=I is the identity operator.In this paper,we establish the sharp Jackson inequality between E(f,■_n) andω_r(f,τ) for some cases of r andτ.More precisely,we will find the smallest constant ■_n(τ,r) which depends only on n,r,andτ,such that the inequality E(f,■_n)≤■_n(τ,r)ω_r(f,τ) is valid.

BEST APPROXIMATION THEOREM FOR SET-VALUED MAPPINGSWITHOUT CONVEX VELUES AND CONTINUITY

In this paper, a new concept of weakly ,convex graph for set-valued mappings is introduced and studied. By using the concept , some new coincidence, the bestapproximation and fixed point-theorems are obtained.

THE ORTHOGONAL POLYNOMIALS AND THE PADE'APPROXIMATION

The diagonal Pade' approximates for exp(x). tanx and tanhx are obtained in asimple manner by using the property of Legendre polynomials that on [ -1, 1] Pn (x)is orthogonal to every polynomial of lower degree. Gauss's quadrature formula is used tofined the denomiators of some functions.

THE BEST APPROXIMATION AND COINCIDENCE THEOREMS FOR COMPOSITES OF ACYCLIC MAPPINGS

Some new coincidence theorem s involving a new class of set_valued mappings containing composites of acyclic mappings defined in a contractible space are proved. For applications, some best approximation theorems and coincidence theorems for set-valued mappings are als o given. A number of known results in recent literature are improved and general ized by the theorems in this paper.

C^p Condition and the Best Local Approximation

In this paper, we introduce a condition weaker than the LP differentiability, which we call Cp condition. We prove that if a function satisfies this condition at a point, then there exists the best local approximation at that point. We also give a necessary and sufficient condition for that a function be LP differentiable. In addition, we study the convexity of the set of cluster points of the net of best appoximations of f, {Pε（f）} asε→0.

A Note on Pad Approximants Pairs as Limits of Algebraic Polynomials Pairs of Weighted Best Approximation in Orlicz Spaces

In this short note, we show the behavior in Orlicz spaces of best approximations by algebraic polynomials pairs on union of neighborhoods, when the measure of them tends to zero.

THE SARD BEST APPROXIMATION OF LINEAR FUNCTIONAL

In numerical analysis, it is significant to approximate the linear functional Ef=sum from i=0 to m-1([integral from a to b(a1（x）f1（x）dx+ sum from f=0 to i1(b1f1（x1）)]) by a simpler linear functional Lf=sum from i=1 to m（a1f（x1）） In this paper, making use of natural Tchebysheff spline function, we give existence theorem and uniqueness theorem of L that is exact for the degree m to F; we also give three sufficient and necessary conditions in which L is the Sard best approximation to F.

A Class of Iteration Method for the Best Approximation Problems

We propose a class of iteration methods searching the best approximately generalized polynomial, which has parallel computational function and converges to the exact solution quadratically. We first transform it into a special system of nonlinear equations with constraint, then by using to certain iteration method, we combine the two basic processes of the Remes method into a whole such that the iterative process of the system of nonlinear equations and the computation of the solution to the system of linear equations proceed alternately. A lot of numerical examples show that this method not only has good convergence property but also always converges to the exact solution of the problem accurately and rapidly for almost all initial approximations .

Maximal Inequalities for the Best Approximation Operator and Simonenko Indices

In an abstract set up, we get strong type inequalities in L^p＋1 by assuming weak or extra-weak inequalities in Orlicz spaces. For some classes of functions, the number p is related to Simonenko indices. We apply the results to get strong inequal- ities for maximal functions associated to best Ф-approximation operators in an Orlicz space L^Ф.

On Copositive Approximation in Spaces of Continuous Functions Ⅱ:The Uniqueness of Best Copositive Approximation

This paper is part II of ＂On Copositive Approximation in Spaces of Contin- uous Functions＂. In this paper, the author shows that if Q is any compact subset of real numbers, and M is any finite dimensional strict Chebyshev subspace of C （Q）, then for any admissible function f ∈ C（Q）／M, the best copositive approximation to f from M is unique.

SOME GENERALIZATIONS OF KY FAN'S BEST APPROXIMATION THEOREM

We pressent new Ky Fan type best approximation theorems for a discontinuous multivalued map on metrizable topological vector spaces and hyperconvex spaces. In addition, fixed point results are derived for the map studied. Our work generalizes severl results in approximation theory.

Pointwise Approximation Theorems for Combinations of Bernstein Polynomials with Inner Singularities

It is well-known that Bernstein polynomials are very important in studying the characters of smoothness in theory of approximation. A new type of combinations of Bernstein operators are given in [1]. In this paper, we give the Bernstein-Markov inequalities with step-weight functions for combinations of Bernstein polynomials with inner singularities as well as direct and inverse theorems.

THE BEST LIPSCHITZ CONSTANTS OF BERNSTEIN POLYNOMIALS AND BEZIER NETS OVER A GIVEN TRIANGLE

This paper proved the following three facts about the Lipschitz continuous property of Bernstein polynomials and Bezier nets defined on a triangle: suppose f(P) is a real valued function defined on a triangle T, (1) If f(P) satisfies Lipschitz continuous condition, i. e. f(P)∈Lip4α, then the corresponding Bernstein Bezier net fn∈LipAsecαψα, here ψ is the half of the largest angle of triangle T; (2) If Bernstein Bezier net fn∈ LipBα, then its elevation Bezier net Efn∈LipBα; and (3) If f(P)∈Lipαa, then the corresponding Bernstein polynomials Bn(f;P)∈LipAsecαψα, and the constant Asecαψ best in some sense.

Some Remarks on the Convex Feasibility Problem and Best Approximation Problem

In this paper we investigate several solution algorithms for the convex fea- sibility problem(CFP)and the best approximation problem(BAP)respectively.The algorithms analyzed are already known before,but by adequately reformulating the CFP or the BAP we naturally deduce the general projection method for the CFP from well-known steepest decent method for unconstrained optimization and we also give a natural strategy of updating weight parameters.In the linear case we show the connec- tion of the two projection algorithms for the CFP and the BAP respectively.In addition, we establish the convergence of a method for the BAP under milder assumptions in the linear case.We also show by examples a Bauschke's conjecture is only partially correct.

AN APPLICATION OF A FIXED POINT THEOREM TO BEST SIMULTANEOUS APPROXIMATION

Using a recent result regarding the fixed points of multivalued mappings, the existence of invariant best simultaneous approximation in chainable metric space is proved.

ON REMES ALGORITHA FOR THE BEST CHEBYSHEV APPROXIMATION FROM VARISOLVENT FAMILY

In this paper, Remes algorithm is applied to compute the numerical solution of the best chebyshev approximation from varisolvent family. Feasibility and convergence of the algorithm are discussed carefully.