Markov—Bernstein-Type Inequality for Trigonometric Polynomials with Respect to Doubling Weights on [-ω,ω]

Abstract. Various important, weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc., have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumptions on the weights. In most cases this minimal assumption is the doubling condition. Here, based on a recently proved Bernstein-type inequality by D. S. Lubinsky, we establish Markov—Bernstein-type inequalities for trigonometric polynomials with respect to doubling weights on [-ω,ω] . Namely, we show the theorem below. Theorem Let p ∈ [1,∞) and ω ∈ (0, 1/2] . Suppose W is a weight function on [-ω,ω] such that W(ω cos t) is a doubling weight. Then there is a constant C depending only on p and the doubling constant L so that $$\smallint _{ - \omega }^\omega \left| {T'_n (t)} \right|^p W(t)(\omega /n + \sqrt {\omega ^2 - t^2 )} ^p dt \leqslant Cn^p \smallint _{ - \omega }^\omega \left| {T_n (t)} \right|^p W(t)dt$$ holds for every T _n ∈ T _n , where T _n denotes the class of all real trigonometric polynomials of degree at most n .     

Zolotarevω-Polynomials inWH[0, 1]

The main result of this paper characterizes generalizationsof Zolotarev polynomials as extremal functions in the Kolmogorov–Landauproblemwhereω(t) is a concave modulus of continuity,r, m: 1?m?r,are integers, andB?B₀(r, m, ω). We show that theextremal functionsZ_Bhaver+1 points of alternance andthe full modulus of continuity ofZ^(r)_B: ω(Z^(r)_B; t)=ω(t) for allt∈[0, 1]. This generalizesthe Karlin's result on the extremality of classical Zolotarevpolynomials in the problem () forω(t)=tand allB?B_r.     

The Boundedness of High Order Riesz-Bessel Transformations Generated by the Generalized Shift Operator in Weighted Lp,ω,γ-spaces with General Weights

In this study, the boundedness of the high order Riesz-Bessel transformations generated by generalized shift operator in weighted L _p,ω,γ -spaces with general weights is proved.     

Optimality Conditions and Duality for?Nondifferentiable Multiobjective Fractional Programming Problems with?(C,α,ρ,d)-convexity

The purpose of this paper is to consider a class of nondifferentiable multiobjective fractional programming problems in which every component of the objective function contains a term involving the support function of a compact convex set. Based on the (C,α,ρ,d)-convexity, sufficient optimality conditions and duality results for weakly efficient solutions of the nondifferentiable multiobjective fractional programming problem are established. The results extend and improve the corresponding results in the literature.     

Asymptotics of Recurrence Relation Coefficients,Hankel Determinant Ratios,and Root Products Associated with Laurent Polynomials Orthogonal with Respect to Varying Exponential Weights

Let Λ^ℝ denote the linear space over ℝ spanned by z ^k , k∈ℤ. Define the real inner product 〈⋅,⋅〉 _L :Λ^ℝ×Λ^ℝ→ℝ, , N∈ℕ, where V satisfies: (i) V is real analytic on ℝ∖{0}; (ii) lim  _{| x |→∞}(V(x)/ln (x ²+1))=+∞; and (iii) lim  _{| x |→0}(V(x)/ln (x ⁻²+1))=+∞. Orthogonalisation of the (ordered) base with respect to 〈⋅,⋅〉 _L yields the even degree and odd degree orthonormal Laurent polynomials (OLPs) : φ _2n (z)=∑ _k=−n ⁿ ξ _k ⁽²ⁿ⁾ z ^k , ξ _n ⁽²ⁿ⁾>0, and φ _2n+1(z)=∑ _k=−n−1 ⁿ ξ _k ⁽²ⁿ⁺¹⁾ z ^k , ξ _−n−1⁽²ⁿ⁺¹⁾>0. Associated with the even degree and odd degree OLPs are the following two pairs of recurrence relations: z φ _2n (z)=c _2n ^♯ φ _2n−2(z)+b _2n ^♯ φ _2n−1(z)+a _2n ^♯ φ _2n (z)+b _2n+1 ^♯ φ _2n+1(z)+c _2n+2 ^♯ φ _2n+2(z) and z φ _2n+1(z)=b _2n+1 ^♯ φ _2n (z)+a _2n+1 ^♯ φ _2n+1(z)+b _2n+2 ^♯ φ _2n+2(z), where c ₀ ^♯ =b ₀ ^♯ =0, and c _2k ^♯ >0, k∈ℕ, and z ⁻¹ φ _2n+1(z)=γ _2n+1 ^♯ φ _2n−1(z)+β _2n+1 ^♯ φ _2n (z)+α _2n+1 ^♯ φ _2n+1(z)+β _2n+2 ^♯ φ _2n+2(z)+γ _2n+3 ^♯ φ _2n+3(z) and z ⁻¹ φ _2n (z)=β _2n ^♯ φ _2n−1(z)+α _2n ^♯ φ _2n (z)+β _2n+1 ^♯ φ _2n+1(z), where β ₀ ^♯ =γ ₁ ^♯ =0, β ₁ ^♯ >0, and γ _2l+1 ^♯ >0, l∈ℕ. Asymptotics in the double-scaling limit N,n→∞ such that N/n=1+o(1) of the coefficients of these two pairs of recurrence relations, Hankel determinant ratios associated with the real-valued, bi-infinite strong moment sequence , and the products of the (real) roots of the OLPs are obtained by formulating the even degree and odd degree OLP problems as matrix Riemann-Hilbert problems on ℝ, and then extracting the large-n behaviours by applying the non-linear steepest-descent method introduced in (Ann. Math. 137(2):295–368, [1993]) and further developed in (Commun. Pure Appl. Math. 48(3):277–337, [1995]) and (Int. Math. Res. Not. 6:285–299, [1997]).      

关于一类弱Berwald的(α,β)-度量

本文研究了一类重要的形如F=α +εβ +β arctan(β/α) (ε为常数)的弱Berwald (α, β)-度量.利用S-曲率公式,获得了这类度量为弱Berwald度量的充要条件.并且还证明了F为具有标量旗曲率的弱Berwald度量当且仅当它们为Berwald度量且旗曲率消失.     

The Rν-generalized solution of a boundary value problem with a singularity belongs to the space W2,ν+β2+κ+1/κ+2 (Ω, δ)

In the present paper, for a boundary value problem with noncoordinated degeneration of the data and a singularity in the solution, we show that the R _ν -generalized solution belongs to the weighted space W _2,ν+gb ^2+κ+1/κ+2 (Ω, δ)(κ > 0). Original Russian Text ? V.A. Rukavishnikov, E.V. Kuznetsova, 2009, published in Differentsial’nye Uravneniya, 2009, Vol. 45, No. 6, pp. 894–898.     

(α,δ)-弱刚性环上的Ore扩张

本文研究（α,δ）-弱刚性环上的Ore扩张环R[x;α,δ]的弱对称性、弱zip性、幂零p.p.性和幂零Baer性.利用对多项式的逐项分析的方法,证明了如果R是（α,δ）-弱刚性环和半交换环,则Ore扩张环R[x;α,δ]是弱对称的（弱zip的,幂零p.p.的,幂零Baer的）当且仅当R是弱对称的（弱zip的,幂零p.p.的,幂零Baer的）.这些结果统一和扩展了前面已有的相关结论.     

Equivalence of Minimal ?0- and ?p-Norm Solutions of Linear Equalities, Inequalities and Linear Programs for Sufficiently Small p

For a bounded system of linear equalities and inequalities, we show that the NP-hard ℓ ₀-norm minimization problem is completely equivalent to the concave ℓ _p -norm minimization problem, for a sufficiently small p. A local solution to the latter problem can be easily obtained by solving a provably finite number of linear programs. Computational results frequently leading to a global solution of the ℓ ₀-minimization problem and often producing sparser solutions than the corresponding ℓ ₁-solution are given. A similar approach applies to finding minimal ℓ ₀-solutions of linear programs.     

Nonexistence of [n, 5, d]q Codes Attaining the Griesmer Bound for q4 ?2q2 ?2q+1 ≤ d≤ q4?2q2 ?q

We prove that there does not exist a [q⁴+q³−q²−3q−1, 5, q⁴−2q²−2q+1]_q code over the finite field for q≥ 5. Using this, we prove that there does not exist a [g_q(5, d), 5, d]_q code with q⁴ −2q² −2q +1 ≤ d ≤ q⁴ −2q² −q for q≥ 5, where g_q(k,d) denotes the Griesmer bound.MSC 2000: 94B65, 94B05, 51E20, 05B25     

Lipschitz conditions for the generalized discrete Fourier transform associated with the Jacobi operator on [0,π]

Our aim in this paper is to prove an analog of the classical Titchmarsh theorem on the image under the discrete Fourier–Jacobi transform of a set of functions satisfying a generalized Lipschitz condition in the space ${\mathbb{L}}_2^{(\alpha ,\beta )}$.     

Generalization of the lions theory for first-order evolution differential equations with discontinuous operators and with α ∈ [1/2, 1]

We prove existence, uniqueness, and smoothness theorems for weak solutions of the problem $$ du(t)/dt + A(t)u(t) = f(t), t \in ]0,T[; u(0) = u_0 \in H, $$ where, for almost all t, the linear unbounded operators A(t) with domains D(A(t)) depending on t are closed and maximal accretive and have bounded inverses A ^?1(t) discontinuous with respect to t in the Hilbert space H. There exists an α ∈ [1/2, 1] such that the following is true in H for almost all t: the power A ^α (t) is subordinate to the power A* ^α (t) of the adjoint operators A*(t), the operators A ^α (t) and A* ^α (t) do not form an obtuse angle, and the domains D(A* ^α (t)) of the operators A* ^α (t) are not increasing with respect to t. This paper is the first to prove the well-posedness of the mixed problem for the multidimensional linearized Korteweg-de Vries equation smooth in time with boundary conditions piecewise constant in time.     

Corrigendum to “Heavy-Traffic Asymptotics for Stationary GI/G/1-Type Markov Chains” [Oper. Res. Lett. 40 (2012) 185–189]

This corrigendum provides corrections for several errors in the previously published letter “Heavy-Traffic Asymptotics for Stationary GI/G/1-Type Markov Chains” [Operations Research Letters 40 (2012) 185–189].     

多目标优化问题(ε,ε)-拟近似解的非线性标量化

本文研究了多目标优化问题的(ε,ε)-拟近似解.利用文献[1]给出的多目标优化问题统一的非线性标量化问题,在没有任何凸性条件下,研究了多目标优化问题的(ε,ε)-拟近似解的充分和必要条件.最后,利用文献[2]中给出的的范数,对多目标优化问题的(ε,ε)-拟近似解进行了非线性标量化刻画.本文第3节推广了文献[1]中的结果.     

The sharp Jackson inequality in the space L2 on the segment [?1, 1] with the power weight

In the space L ₂ on the segment [−1, 1] with the power weight |x|^2λ+1, λ ≥ −1/2, we define a complete orthogonal system, the value of the best approximation with respect to this system, the operator of generalized shift, and the modulus of continuity and prove the sharp Jackson inequality.     

Erratum to “The Weighted Estimates of the SchrSdinger Operators on the Nilpotent Lie Group” [J.Math.Res.Exposition, 2010, 30(6): 1117-1124]

Refers to:The Weighted Estimates of the Schr(o)dinger Operators on the Nilpotent Lie Group Journal of Mathematical Research and Exposition,Volume 30,Issue 6,20November 2010,Pages 1117-1124Ling Juan HAN and Qiao Zhen MA     

Optimal regularity for [`(?)] b\bar \partial _b on CR manifoldson CR manifolds

In this paper a new explicit integral formula is derived for solutions of the tangential Cauchy-Riemann equations on CR q-concave manifolds and optimal estimates in the Lipschitz norms are obtained.