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→∞.     

Generalized Bernstein Polynomials

We introduce polynomials B ⁿ _i (x;ω|q), depending on two parameters q and ω, which generalize classical Bernstein polynomials, discrete Bernstein polynomials defined by Sablonnière, as well as q-Bernstein polynomials introduced by Phillips. Basic properties of the new polynomials are given. Also, formulas relating B ⁿ _i (x;ω|q), big q-Jacobi and q-Hahn (or dual q-Hahn) polynomials are presented. This revised version was published online in July 2006 with corrections to the Cover Date.     

Voronovskaya-type formulas and saturation of convergence for q-Bernstein polynomials for

In the paper, we discuss Voronovskaya-type theorem and saturation of convergence for q-Bernstein polynomials for arbitrary fixed q, 0<q<1. We give explicit formulas of Voronovskaya-type for the q-Bernstein polynomials for 0<q<1. If , we show that the rate of convergence for the q-Bernstein polynomials is o(qⁿ) if and only ifWe also prove that if f is convex on [0,1] or analytic on (-ε,1+ε) for some ε>0, then the rate of convergence for the q-Bernstein polynomials is o(qⁿ) if and only if f is linear.     

The sharpness of convergence results for <Emphasis Type="Italic">q</Emphasis>-Bernstein polynomials in the case <Emphasis Type="Italic">q</Emphasis> > 1

Due to the fact that in the case q > 1 the q-Bernstein polynomials are no longer positive linear operators on C[0, 1], the study of their convergence properties turns out to be essentially more difficult than that for q < 1. In this paper, new saturation theorems related to the convergence of q-Bernstein polynomials in the case q > 1 are proved.     

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.     

The approximation by q-Bernstein polynomials in the case q ↓ 1

Let B_n (f, q; x), n=1, 2, ... , 0 < q < ∞, be the q-Bernstein polynomials of a function f, B_n (f, 1; x) being the classical Bernstein polynomials. It is proved that, in general, {B_n (f, q_n; x)} with q_n ↓ 1 is not an approximating sequence for f ∈C[0, 1], in contrast to the standard case q_n ↓ 1. At the same time, there exists a sequence 0 < δ_n ↓ 0 such that the condition implies the approximation of f by {B_n (f, q_n; x)} for all f ∈C[0, 1]. Received: 15 March 2005     

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].     

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.     

Tensor product q-Bernstein polynomials

An affine de Casteljau type algorithm to compute q-Bernstein Bézier curves is introduced and its intermediate points are obtained explicitly in two ways. Furthermore we define a tensor product patch, based on this algorithm, depending on two parameters. Degree elevation procedure is studied. The matrix representation of tensor product patch is given and we find the transformation matrix between a classical tensor product Bézier patch and a tensor product q-Bernstein Bézier patch. Finally, q-Bernstein polynomials B _n,m (f;x,y) for a function f(x,y), (x,y)∈[0,1]×[0,1] are defined and fundamental properties are discussed. AMS subject classification (2000)  65D17     

Best m-ter m approxi mation of the classes$$ B_{\infty, \theta }^r $$ of functions of many variables by polyno mials in the haar syste m

We obtain an exact-order estimate for the best m-term approximation of the classes B_{￥, q}^r B_{\infty, \theta }^r of periodic functions of many variables by polynomials in the Haar system in the metric of the space L _q , 1 < q < ∞.     

Positive linear operators generated by analytic functions

Let φ be a power series with positive Taylor coefficients {a _k } _k=0^∞ and non-zero radius of convergence r ≤ ∞. Let ξ _x , 0 ≤ x < r be a random variable whose values α _k , k = 0, 1, …, are independent of x and taken with probabilities a _k x ^k /φ(x), k = 0, 1, …. The positive linear operator (A _φ f)(x):= E[f(ξ _x )] is studied. It is proved that if E(ξ _x ) = x, E(ξ _x ²) = qx ² + bx + c, q, b, c ∈ R, q > 0, then A _φ reduces to the Szász-Mirakyan operator in the case q = 1, to the limit q-Bernstein operator in the case 0 < q < 1, and to a modification of the Lupaş operator in the case q > 1.     

An acyclic extension of the braid group

We relate Artin's braid groupB _℞=lim_→B_n to a certain groupF′ ofpl-homeomorphisms of the interval. Namely, there exists a short exact sequence 1→B _∞→A→F′→1 whereH _kA=0,k≥1.     

The approximation of logarithmic function by q-Bernstein polynomials in the case q > 1

Since in the case q > 1, q-Bernstein polynomials are not positive linear operators on C[0,1], the study of their approximation properties is essentially more difficult than that for 0<q<1. Despite the intensive research conducted in the area lately, the problem of describing the class of functions in C[0,1] uniformly approximated by their q-Bernstein polynomials (q > 1) remains open. It is known that the approximation occurs for functions admit ting an analytic continuation into a disc {z:|z| < R}, R > 1. For functions without such an assumption, no general results on approximation are available. In this paper, it is shown that the function f(x) = ln (x + a), a > 0, is uniformly approximated by its q-Bernstein polynomials (q > 1) on the interval [0,1] if and only if a ≥ 1.      

On the order of relative approximation of classes of differentiable periodic functions by splines

In the case where n → ∞, we obtain order equalities for the best L _q -approximations of the classes W _p ^r , 1 ≤ q ≤ p ≤ 2, of differentiable periodical functions by splines from these classes.     

Approximative characteristics of the isotropic classes of periodic functions of many variables

Exact-order estimates are obtained for the best orthogonal trigonometric approximations of the Besov (B _p,θ ^r ) and Nukol’skii (H _p ^r ) classes of periodic functions of many variables in the metric of L _q , 1 ≤ p, q ≤ ∞. We also establish the orders of the best approximations of functions from the same classes in the spaces L ₁ and L _∞ by trigonometric polynomials with the corresponding spectrum.     

Bounded imaginary powers and <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript>-calculus of the Stokes operator in two-dimensional exterior domains

The present contribution deals with the Stokes operator A_q on L^q_σ(Ω), 1<q<∞, where Ω is an exterior domain in ℝ² of class C². It is proved that A_q admits a bounded H_∞-calculus. This implies the existence of bounded imaginary powers of A_q, which has several important applications. – So far this property was only known for exterior domains in ℝⁿ, n≥3. – In particular, this shows that A_q has maximal regularity on L^q_σ(Ω). For the proof the resolvent (λ+A_q)⁻¹ has to be analyzed for |λ|→∞ and λ→0. For large λ this is done using an approximate resolvent based on the results of [3], which were obtained by applying the calculus of pseudodifferential boundary value problems. For small λ we analyze the representation of the resolvent developed in [11] by a potential theoretical method.     

Asymptotic formulas for the enumerator of trees with a given number of hanging or internal vertices

Let t(r, n) be the number of trees with n vertices of which r are hanging and q are internal (r=n−9). For a fixed r or q we prove the validity of the asymptotic formulas (r > 2)t(r, n)≈t/r|(r−2)| 2^2−r n ^2r−4 (n→∞)t(n−q, n)≈1/q|(q&#x2212;1)|q ^q−2 n ^q−1 (n→∞) In the derivation of these formulas we do not use the expression for the enumerator of the trees with respect to the number of hanging vertices. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 65–70, January, 1977.     

Best approximation by ridge functions in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We study the approximation of the classes of functions by the manifold R _n formed by all possible linear combinations of n ridge functions of the form r(a · x)): It is proved that, for any 1 ≤ q ≤ p ≤ ∞, the deviation of the Sobolev class W ^r _p from the set R _n of ridge functions in the space L _q (B ^d ) satisfies the sharp order n ^-r/(d-1).     

Almost Everywhere Convergence of Fejér Means of <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> Functions on Rarely Unbounded Vilenkin Groups

A highly celebrated problem in dyadic harmonic analysis is the pointwise convergence of the Fejér (or (C, 1)) means of functions on unbounded Vilenkin groups. There are several papers of the author of this paper concerning this. That is, we know the a.e. convergence σ _n f → f (n → ∞) for functions f ∈ L ^p , where p > 1 (Journal of Approximation Theory, 101(1), 1–36, (1999)) and also the a.e. convergence σM _n f → f (n → ∞) for functions f ∈ L ¹ (Journal of Approximation Theory, 124(1), 25–43, (2003)). The aim of this paper is to prove the a.e. relation lim _n → σ _n f = f for each integrable function f on any rarely unbounded Vilenkin group. The concept of the rarely unbounded Vilenkin group is discussed in the paper. Basically, it means that the generating sequence m may be an unbounded one, but its "big elements" are not "too dense". Research supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. M 36511/2001 and T 048780     

An answer to a question by J. Milnor

We consider two commuting automorphismsT ₁,T ₂ of the Lebesque space (M, M, μ) such thath _m,n=h(T ₁ ^m T ₂ ⁿ )<∞ whereh is the measure-theoretic entropy. Under additional assumptions we show the existence of the limits lim (1/m)h _m,n wherem→∞,n→∞,m/n→ω and ω is an irrational number.