The rate of convergence of q-Bernstein polynomials for

In the note, we obtain the estimates for the rate of convergence for a sequence of q-Bernstein polynomials {B_n,q(f)} for 0<q<1 by the modulus of continuity of f, and the estimates are sharp with respect to the order for Lipschitz continuous functions. We also get the exact orders of convergence for a family of functions , and the orders do not depend on α, unlike the classical case.     

Zero-density estimates for automorphic <Emphasis Type="Italic">L</Emphasis>-functions

In this paper we prove zero-density estimates of the large sieve type for the automorphic L-function L（s, f × χ）, where f is a holomorphic cusp form and χ（mod q） is a primitive character.     

Properties of convergence for -Bernstein polynomials

In this paper, we discuss properties of the ω,q-Bernstein polynomials introduced by S. Lewanowicz and P. Woźny in [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT 44 (1) (2004) 63–78], where fC[0,1], ω,q>0, ω≠1,q⁻¹,…,q⁻ⁿ⁺¹. When ω=0, we recover the q-Bernstein polynomials introduced by [G.M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511–518]; when q=1, we recover the classical Bernstein polynomials. We compute the second moment of , and demonstrate that if f is convex and ω,q(0,1) or (1,∞), then are monotonically decreasing in n for all x[0,1]. We prove that for ω(0,1), q_n(0,1], the sequence converges to f uniformly on [0,1] for each fC[0,1] if and only if lim_n→∞q_n=1. For fixed ω,q(0,1), we prove that the sequence converges for each fC[0,1] and obtain the estimates for the rate of convergence of by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions.     

The minimizing of the Nielsen root classes

Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen root number. The condition “f satisfies the Wecken property is known to be equivalent to |deg(f)|≤N R[f]/(1−χ(M ₂)−χ(M ₁0/(1−χ(M ₂)) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f)≤N R[f]/(1−χ(M ₂)−χ(M ₂)/(1−χ(M ₂)). Also we construct, for each integer n≥3, an example of a map f: K _n →N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time.     

Transcendence measures and algebraic growth of entire functions

In this paper we obtain estimates for certain transcendence measures of an entire function f. Using these estimates, we prove Bernstein, doubling and Markov inequalities for a polynomial P(z,w) in ℂ² along the graph of f. These inequalities provide, in turn, estimates for the number of zeros of the function P(z,f(z)) in the disk of radius r, in terms of the degree of P and of r. Our estimates hold for arbitrary entire functions f of finite order, and for a subsequence {n _j } of degrees of polynomials. But for special classes of functions, including the Riemann ζ-function, they hold for all degrees and are asymptotically best possible. From this theory we derive lower estimates for a certain algebraic measure of a set of values f(E), in terms of the size of the set E.     

On the Convergence and Iterates of q-Bernstein Polynomials

The convergence properties of q-Bernstein polynomials are investigated. When q1 is fixed the generalized Bernstein polynomials _nf of f, a one parameter family of Bernstein polynomials, converge to f as n→∞ if f is a polynomial. It is proved that, if the parameter 0<q<1 is fixed, then _nf→f if and only if f is linear. The iterates of _nf are also considered. It is shown that _n^Mf converges to the linear interpolating polynomial for f at the endpoints of [0,1], for any fixed q>0, as the number of iterates M→∞. Moreover, the iterates of the Boolean sum of _nf converge to the interpolating polynomial for f at n+1 geometrically spaced nodes on [0,1].     

q-Bernstein polynomials and their iterates

Let B_n( f,q;x), n=1,2,… be q-Bernstein polynomials of a function f : [0,1]→C. The polynomials B_n( f,1;x) are classical Bernstein polynomials. For q≠1 the properties of q-Bernstein polynomials differ essentially from those in the classical case. This paper deals with approximating properties of q-Bernstein polynomials in the case q>1 with respect to both n and q. Some estimates on the rate of convergence are given. In particular, it is proved that for a function f analytic in {z: |z|<q+} the rate of convergence of {B_n( f,q;x)} to f(x) in the norm of C[0,1] has the order q⁻ⁿ (versus 1/n for the classical Bernstein polynomials). Also iterates of q-Bernstein polynomials {B_n^j_n( f,q;x)}, where both n→∞ and j_n→∞, are studied. It is shown that for q(0,1) the asymptotic behavior of such iterates is quite different from the classical case. In particular, the limit does not depend on the rate of j_n→∞.     

<Emphasis Type="Italic">f</Emphasis>-Colorings of some graphs of <Emphasis Type="Italic">f</Emphasis>-class 1

An f-coloring of a graph G is an edge-coloring of G such that each color appears at each vertex v V（G） at most f（v） times. The minimum number of colors needed to f-color G is called the f-chromatic index of G and is denoted by X′f（G）. Any simple graph G has the f-chromatic index equal to △f（G） or △f（G） ＋ 1, where △f（G） =max v V（G）{[d（v）/f（v）]}. If X′f（G） = △f（G）, then G is of f-class 1; otherwise G is of f-class 2. In this paper, a class of graphs of f-class 1 are obtained by a constructive proof. As a result, f-colorings of these graphs with △f（G） colors are given.     

Structural Remarks on Bipartite Graphs with Unique f-Factors

In this note we will derive some structural results for a bipartite graph G with a unique f-factor. Two necessary conditions will be that G is saturated, meaning that the addition of any edge leads to a second f-factor, and that f_A, f_B≥1. Here f_A and f_B are defined as the minimum of f over the vertices in the two partite sets A and B of G, respectively. Our main result states that G has at least f_A + f_B vertices for which d_G (v) = f(v) holds.     

Some limit results for probabilities estimates of Brownian motion with polynomial drift

Let (B _t + f(t)) _t∈[0,+∞) be a Brownian motion with polynomial drift f(t), where f(t) is a polynomial. Some Limit Results for Lower tail and large deviation probabilities estimates, and Level crossing probabilities estimates of (B _t + f(t)) _t∈[0,+∞) are given in this paper.     

On lipschitz continuity of quasiconformalmappings in space

Sufficient conditions for a spatial quasiconformal mapping f to be Lipschitz or weak Lipschitz continuous at a prescribed point are given. The proofs of these results are based on new growth estimates for the conformal modulus under quasiconformal mappings f. The estimates are written in terms of integral means of the co-called pointwise angular dilatation coefficient.     

Inequalities involving independence domination, f-domination, connected and total f-domination numbers

Let f be an integer-valued function defined on the vertex set V(G) of a graph G. A subset D of V(G) is an f-dominating set if each vertex x outside D is adjacent to at least f(x) vertices in D. The minimum number of vertices in an f-dominating set is defined to be the f-domination number, denoted by _f (G). In a similar way one can define the connected and total f-domination numbers _c,f (G) and _t,f (G). If f(x) = 1 for all vertices x, then these are the ordinary domination number, connected domination number and total domination number of G, respectively. In this paper we prove some inequalities involving _f (G), _c,f (G), _t,f (G) and the independence domination number i(G). In particular, several known results are generalized.     

On coefficient estimates for a class of holomorphic mappings

Let X be a complex Banach space with norm ‖ · ‖, B be the unit ball in X, D ⁿ be the unit polydisc in ℂ ⁿ . In this paper, we introduce a class of holomorphic mappings on B or D ⁿ . Let f(x) be a normalized locally biholomorphic mapping on B such that (Df(x))⁻¹ f(x) ∈ and f(x) − x has a zero of order k + 1 at x = 0. We obtain coefficient estimates for f(x). These results unify and generalize many known results. This work was supported by National Natural Science Foundation of China (Grant No. 10571164), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20050358052), the Jiangxi Provincial Natural Science Foundation of China (Grant No. 2007GZS0177) and Specialized Research Fund for the Doctoral Program of Jiangxi Normal University.     

Sharp regularity results on second derivatives of solutions to the Monge‐Ampère equation with VMO type data

We establish interior estimates for L^p‐norms, Orlicz norms, and mean oscillation of second derivatives of solutions to the Monge‐Ampère equation det D²u = f(x) with zero boundary value, where f(x) is strictly positive, bounded, and satisfies a VMO‐type condition. These estimates develop the regularity theory of the Monge‐Ampère equation in VMO‐type spaces. Our Orlicz estimates also sharpen Caffarelli's celebrated W^{2, p}‐estimates. © 2008 Wiley Periodicals, Inc.     

On the Rates of Approximation of Bernstein Type Operators

Asymptotic behavior of two Bernstein-type operators is studied in this paper. In the first case, the rate of convergence of a Bernstein operator for a bounded function f is studied at points x where f(x+) and f(x−) exist. In the second case, the rate of convergence of a Szász operator for a function f whose derivative is of bounded variation is studied at points x where f(x+) and f(x−) exist. Estimates of the rate of convergence are obtained for both cases and the estimates are the best possible for continuous points.     

On regular polynomial endomorphisms of ℂ<Superscript>2</Superscript> without bounded critical orbitswithout bounded critical orbits

We study conditions involving the critical set of a regular polynomial endomorphism f∶ℂ²↦ℂ² under which all complete external rays from infinity for f have well defined endpoints.     

Almost everywhere convergence of Banach space-valued Vilenkin-Fourier series

The duality between martingale Hardy and BMO spaces is generalized for Banach space valued martingales. It is proved that if X is a UMD Banach space and f ∈ L _p(X) for some 1 < p < ∞ then the Vilenkin-Fourier series of f converges to f almost everywhere in X norm, which is the extension of Carleson’s result. This paper was written while the author was researching at University of Vienna (NuHAG) supported by Lise Meitner fellowship No. M733-N04. This research was also supported by the Hungarian Scientific Research Funds (OTKA) No. T043769, T047128, T047132.     

β级星像函数类和凸像函数类的扩展问题

本文利用文[2,3]的引理和算子L(a,c)f(z)的一些性质.结合Hadamard乘积,研究了算子L(a,c)f(z),获得了L(a,c)f(z)∈S*(β)和L(a,c)f(z)∈K(β)的充分条件,推广了文[2,3]的相关结论.     

On the Nevanlinna characteristic of <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">z</Emphasis>+<Emphasis Type="Italic">η</Emphasis>) and difference equations in the complex plane

We investigate the growth of the Nevanlinna characteristic of f(z+η) for a fixed η∈C in this paper. In particular, we obtain a precise asymptotic relation between T(r,f(z+η)) and T(r,f), which is only true for finite order meromorphic functions. We have also obtained the proximity function and pointwise estimates of f(z+η)/f(z) which is a discrete version of the classical logarithmic derivative estimates of f(z). We apply these results to give new growth estimates of meromorphic solutions to higher order linear difference equations. This also allows us to solve an old problem of Whittaker (Interpolatory Function Theory, Cambridge University Press, Cambridge, 1935) concerning a first order difference equation. We show by giving a number of examples that all of our results are best possible in certain senses. Finally, we give a direct proof of a result in Ablowitz, Halburd and Herbst (Nonlinearity 13:889–905, 2000) concerning integrable difference equations. This research was supported in part by the Research Grants Council of the Hong Kong Special Administrative Region, China (HKUST6135/01P). The second author was also partially supported by the National Natural Science Foundation of China (Grant No. 10501044) and the HKUST PDF Matching Fund.