ON THE SATURATION OF L_w~p-APPROXIMATION BY (O-q'-q) TYPE HERMITE-FEJER INTERPOLATING POLYNOMIALS

The "o" saturation theorem and the degree of Lwp, approximation by (0 - q' - q) type Hermite-Fejer interpolating polynomials for mean convergence are obtained.     

ON THE SATURATION OF Lw^p-APPROXIMATIONBY （O-q＇-q） TYPE HERMITE-FEJER INTERPOLATING POLYNOMIALS

The “o” saturation theorem and the degree of L^p _w approximation by (0−q′−q) type Hermite-Fejér interpolating polynomials for mean convergence are obtained. This work is supported by the Doctor Foundation (No:02J20102-06) and the Post Doctor Foundation of Ningbo University.     

Weak Copositive and Intertwining Approximation

It is known that shape preserving approximation has lower rates than unconstrained approximation. This is especially true for copositive and intertwining approximations. ForfL_p, 1p<∞, the former only has rateω(f, n⁻¹)_p, and the latter cannot even be bounded byC f_p. In this paper, we discuss various ways to relax the restrictions in these approximations and conclude that the most sensible way is the so-calledalmostcopositive/intertwining approximation in which one relaxes the restriction on the approximants in a neighborhood of radiusΔ_n(y_j) of each sign changey_j.     

Nearly monotone spline approximation in

It is shown that the rate of -approximation of a non-decreasing function in , , by ``nearly non-decreasing" splines can be estimated in terms of the third classical modulus of smoothness (for uniformly spaced knots) and third Ditzian-Totik modulus (for Chebyshev knots), and that estimates in terms of higher moduli are impossible. It is known that these estimates are no longer true for ``purely" monotone spline approximation, and properties of intervals where the monotonicity restriction can be relaxed in order to achieve better approximation rate are investigated.

     

Equivalence of a -functional with the approximation behavior of some linear means for abstract Fourier series

Within the setting of abstract Cesàro-bounded Fourier series a -functional is introduced and characterized by the convergence behavior of some linear means. Applications are given within the framework of Jacobi, Laguerre and Hermite expansions. In particular, Ditzian's (1996) equivalence result in the setting of Legendre expansions is covered.

     

On Equivalence of Moduli of Smoothness

It is known that iffW^k_p, thenω_m(f, t)_ptω_m−1(f′, t)_p…. Its inverse with any constants independent offis not true in general. Hu and Yu proved that the inverse holds true for splinesSwith equally spaced knots, thusω_m(S, t)_ptω_m−1(S′, t)_pt²ω_m−2(S″, t)_p…. In this paper, we extend their results to splines with any given knot sequence, and further to principal shift-invariant spaces and wavelets under certain conditions. Applications are given at the end of the paper.     

Univariate splines: Equivalence of moduli of smoothness and applications

Several results on equivalence of moduli of smoothness of univariate splines are obtained. For example, it is shown that, for any , , and , the inequality , , is satisfied, where is a piecewise polynomial of degree on a quasi-uniform (i.e., the ratio of lengths of the largest and the smallest intervals is bounded by a constant) partition of an interval. Similar results for Chebyshev partitions and weighted Ditzian-Totik moduli of smoothness are also obtained. These results yield simple new constructions and allow considerable simplification of various known proofs in the area of constrained approximation by polynomials and splines.

     

On approximation by rational functions with prescribed numerator degree in L p spaces

Summary It is proved that, if <InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>f(x)\in L^p_{[-1,1]}$, $1< p\ki \infty$, changes sign exactly $l$ times, then there exists a real rational function $r(x)\in R_{n}^l$ such that <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> {\|f-r\|}_{p}\le C_{p,\delta}{(l+1)}^2\omega {(f,n^{-1})}_p, $$ which generalizes a result of Leviatan and Lubinsky in \cite{4}. A weaker similar result in $L^1_{[-1,1]}$ is also established.     

(2,1)-Total labelling of outerplanar graphs

The (2,1)-total labelling number of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G such that no two adjacent vertices have the same label, no two adjacent edges have the same label and the difference between the labels of a vertex and its incident edges is at least 2. In this paper we prove that if G is an outerplanar graph with maximum degree Δ(G), then if Δ(G)?5, or Δ(G)=3 and G is 2-connected, or Δ(G)=4 and G contains no intersecting triangles.     

Determination of msd(Ln)

The median stabilization degree (msd, for short) of a median algebra measures the largest possible number of steps needed to generate a subalgebra with an arbitrary set of generators. With computer assistance, we found that msd of the lattice {– 1, 0, 1 }4 equals 2. This value is of critical importance to determine msd of {– 1, 0, 1}n for all n 5 and to determine msd of the free median algebra (r) for almost all r 5.     

Some properties of rational approximations of degree (k, 1) in the hardy spaceH 2(D)

We prove that the well-known interpolation conditions for rational approximations with free poles are not sufficient for finding a rational function of the least deviation. For rational approximations of degree (k, 1), we establish that these interpolation conditions are equivalent to the assertion that the interpolation pointc is a stationary point of the functionk(c) defined as the squared deviation off from the subspace of rational functions with numerator of degree k and with a given pole 1/¯c. For any positive integersk ands, we construct a functiong H₂(D) such thatR _k ,1(g)=R _k +s,1(g) > 0. whereR _k ,1(g) is the least deviation ofg from the class of rational function of degree (k, 1).Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 251–259, August, 1998.The author is keenly grateful to N. S. Vyacheslavov, E. P. Dolzhenko, and V. G. Zinov for useful discussions.     

Coconvex approximation in the uniform norm: the final frontier

Summary The paper deals with approximation of a continuous function, on a finite interval, which changes convexity finitely many times, by algebraic polynomials which are coconvex with it. We give final answers to open questions concerning the validity of Jackson type estimates involving the weighted Ditzian-Totik moduli of smoothness.     

加权范数下Bernstein－Durrmeyer多项式的一个饱和结果

研究Ｂｅｒｎｓｔｅｉｎ－Ｄｕｒｒｍｅｙｅｒ多项式的加权逼近并建立其饱和定理．     

基于Laguerre节点系Hermite-Fejér插值逼近的精确阶

For the nonpositive Hermite-Fejér interpolation based on the Laguerre abscissas, a pointwise two-sided estimate of the degree of approximation in the aleatoric interval [0, A] is first established.     

Periodic Groups Saturated with the Groups <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

Given an indexing set I and a finite field K_α for each α ∈ I, let ? = {L₂(K_α) | α ∈ I} and \(\mathfrak{N} = \{ SL_2 (K_\alpha )|\alpha \in I\}\). We prove that each periodic group G saturated with groups in \(\Re (\mathfrak{N})\) is isomorphic to L₂(P) (respectively SL₂(P)) for a suitable locally finite field P.