A markov type inequality for higher derivatives of polynomials

Let ‖·‖ be the weightedL ²-norm with weightw(t). LetP _n be the set of all complex polynomials whose degree does not exceedn and let \(\gamma _n^{(r)} : = \sup _{f \in P_n } \) (‖f ^(r)‖/‖f‖). In this paper we given upper and lower bound for γ _n ^(r) in the case of the Laguerre weight functionw(t)=exp (?t) and investigate its behaviour asn→∞. Moreover, we derive some identities concerningorthogonal polynomials.     

A grothendieck factorization theorem on 2-convex schatten spaces

We prove that for every bounded linear operatorT:C ^2p →H(1≤p<∞,H is a Hilbert space,C ^{2 p} p is the Schatten space) there exists a continuous linear formf onC ^p such thatf≥0, ‖f‖(C _C ^p)*=1 and $$\forall x \in C^{2p} , \left\| {T(x)} \right\| \leqslant 2\sqrt 2 \left\| T \right\|< f\frac{{x * x + xx*}}{2} > 1/2$$ . Forp=∞ this non-commutative analogue of Grothendieck’s theorem was first proved by G. Pisier. In the above statement the Schatten spaceC ^2p can be replaced byE _E ² whereE ⁽²⁾ is the 2-convexification of the symmetric sequence spaceE, andf is a continuous linear form onC _E. The statement can also be extended toL _E{(su2)}(M, τ) whereM is a Von Neumann algebra,τ a trace onM, E a symmetric function space.     

ДИФФЕРЕНцИАльНыЕ сВ ОИстВА ИжВИВАУЩИхсь ФУНкцИИ

A continuous real valued function defined on an intervalD is called crinkly iff the setf ^?1(У) ∩I is uncountable for each interval \(I \subseteqq D\) and number \(y \in (\mathop {\inf }\limits_I f,\mathop {\sup }\limits_I f)\) . The main result of the paper consists in the following assertion. Let the closed segment [0, 1] be represented as a union of four measurable, mutually nonintersecting setsE ₁,Е ₂,E ₃,E ₄. Then, for each functionH(δ) such thatH(δ)→ + ∞ andδH(δ)→0 asδ→0, there exists a crinkly functionf possessing the following five properties:

1. a.e. onE ₁:D ⁺ f(x)=D-f(x)=+∞,D ₊ f(x)=D?f(x)=?∞;
2. a.e. onE ₂:D ⁺ f(x)=+∞,D?f(x)=?∞,D ₊f(x)=D-f(x)=0;
3. a.e. onE ₃:D ₊ f(x)=?∞,D ^? f(x)=+∞,D ⁺ f(x)=D?f(x)=0;
4. a.e. onE ₄:Df(x)=0;
5. the modulus of continuityΩ off on [0, 1] satisfies $$\omega (\delta ,f,[0,1]) \leqq \delta H(\delta ).$$

     

Potential theory associated with Uhlenbeck-Ornstein process

Some parallel results of Gross' paper (Potential theory on Hilbert space, J. Functional Analysis1 (1967), 123–181) are obtained for Uhlenbeck-Ornstein process U(t) in an abstract Wiener space (H, B, i). Generalized number operator $\text{N}$ is defined by $\text{N}$f(x) = ?lim_∈←0{E[f($\text{U}$(τ^∈_ξ))] ? f(x)}/E[τ^∈_ξ, where τ_x^? is the first exit time of U(t) starting at x from the ball of radius ? with center x. It is shown that $\text{N}$f(x) = ?trace D²f(x)+〈Df(x),x〉 for a large class of functions f. Let r_t(x, dy) be the transition probabilities of U(t). The λ-potential G_λf, λ > 0, and normalized potential Rf of f are defined by G_λf(X) = ∫₀^∞e^?λtr_tf(x) dt and Rf(x) = ∫₀^∞ [r_tf(x) ? r_tf(0)] dt. It is shown that if f is a bounded Lip-1 function then trace D²G_λf(x) ? 〈DG_λf(x), x〉 = ?f(x) + λG_λf(x) and trace D²Rf(x) ? 〈DRf(x), x〉 = ?f(x) + ∫_Bf(y)p₁(dy), where p₁ is the Wiener measure in B with parameter 1. Some approximation theorems are also proved.     

On an interpolatory product rule for evaluating Cauchy principal value integrals

Convergence results for interpolatory product rules for evaluating Cauchy principal value integrals of the form f _?1 ¹ v(x)f(x)/x ? λ dx wherev is an admissible weight function have been extended to integrals of the form f _?1 ¹ k(x)f(x)/x ? λ dx wherek is an arbitrary integrable function subject to certain conditions. Further, whereas the above convergence results were shown when the interpolation points were the Gauss points with respect to some admissible weight functionw, they are now shown to hold when the interpolation points are Radau or Lobatto points with respect tow.     

On the number of positive solutions of elliptic systems

The paper deals with the existence, multiplicity and nonexistence of positive radial solutions for the elliptic system div(|?|^{p –2}?) + λk_i (|x |) fⁱ (u₁, …,u_n) = 0, p > 1, R₁ < |x | < R₂, u_i (x) = 0, on |x | = R₁ and R₂, i = 1, …, n, x ∈ ?^N , where k_i and fⁱ, i = 1, …, n, are continuous and nonnegative functions. Let u = (u₁, …, u_n), φ (t) = |t |^{p –2}t, fⁱ₀ = lim_{‖ u ‖→0}((fⁱ ( u ))/(φ (‖ u ‖))), fⁱ_∞= lim_{‖ u ‖→∞}((fⁱ ( u ))/(φ (‖ u ‖))), i = 1, …, n, f = (f¹, …, fⁿ), f ₀ = ∑ⁿ _{i =1} fⁱ ₀ and f _∞ = ∑ⁿ _{i =1} fⁱ _∞. We prove that either f ₀ = 0 and f _∞ = ∞ (superlinear), or f ₀ = ∞and f _∞ = 0 (sublinear), guarantee existence for all λ > 0. In addition, if fⁱ ( u ) > 0 for ‖ u ‖ > 0, i = 1, …, n, then either f ₀ = f _∞ = 0, or f ₀ = f _∞ = ∞, guarantee multiplicity for sufficiently large, or small λ, respectively. On the other hand, either f₀ and f_∞ > 0, or f₀ and f_∞ < ∞ imply nonexistence for sufficiently large, or small λ, respectively. Furthermore, all the results are valid for Dirichlet/Neumann boundary conditions. We shall use fixed point theorems in a cone. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sharp bounds for integral means

We introduce a bound M of f, ‖f‖_∞?M?2‖f‖_∞, which allows us to give for 0?p<∞ sharp upper bounds, and for −∞<p<0 sharp lower bounds for the average of |f|^p over E if the average of f over E is zero. As an application we give a new proof of Grüss's inequality estimating the covariance of two random variables. We also give a new estimate for the error term in the trapezoidal rule.     

Local properties of functions and approximation by trigonometric polynomials

Suppose Φ_{p, E} (p>0 an integer, E ?[0, 2π]) is a family of positive nondecreasing functions? _x(t) (t>0, x∈ E) such that? _x(nt)≤n^P ? _x(t) (n=0,1,...), t_n is a trigonometric polynomial of order at most n, and Δ _h ^l (f, x) (l>0 an integer) is the finite difference of orderl with step h of the functionf.THEOREM. Supposef (x) is a function which is measurable, finite almost everywhere on [0, 2π], and integrable in some neighborhood of each point xε E,? _X εΦ_p,E and $$\overline {\mathop {\lim }\limits_{\delta \to \infty } } |(2\delta )^{ - 1} \smallint _{ - \delta }^\delta \Delta _u^l (f,x)du|\varphi _x^{ - 1} (\delta ) \leqslant C(x)< \infty (x \in E).$$ . Then there exists a sequence {t _n } _n=1 ^∞ which converges tof (x) almost everywhere, such that for x ε E $$\overline {\mathop {\lim }\limits_{n \to \infty } } |f(x) - l_n (x)|\varphi _x^{ - 1} (l/n) \leqslant AC(x),$$ where A depends on p andl.     

Asymptotics for transportation cost in high dimensions

LetX ₁,...,X _n ,Y ₁,...,Y _n be i.i.d. with the law μ on the cube [0, 1] ^d ,d?3. LetL _n (μ)=inf_πΣ _i=1 ⁿ ||X _i ?Y _π(i)|| denote the optimal bipartite matching of theX andY points, where π ranges over all permutations of the integers 1, 2,...,n, and where ‖·‖ is a norm on ? ^d . If μ is Lebesgue measure it is shown that $$\mathop {\lim }\limits_{n \to \infty } L_n (\mu )/n^{(d - 1)/d} = \alpha {\text{a}}{\text{.s}}{\text{.}}$$ where α is a finite constant depending on ‖ ‖ andd only. More generally, for arbitrary μ it is shown that $$\mathop {\lim }\limits_{n \to \infty } L_n (\mu )/n^{(d - 1)/d} = \alpha \int {(f{\text{(}}x{\text{)}})^{(d - 1)/d} dxa.s.} $$ wheref is the density of the absolutely continuous part of μ. We also find the rate of convergence.     

Sulla misurabilita' delle funzioni di piu' variabili

LetE be a Lebesgue measurable set in IR ^p+q andY a metric space. Iff:E→Y is such thatf(.,x) isL-measurable for almost allx andf(t,.) is continuous in each of theq variables separately for almost allt, thenf must beL-measurable (Theorem 1). By this result we deduce that a functionf:E→Y is almost-continuous iff it is almost-separately continuous. Finally, we give another characterization of the measurability of a functionf:IRp+q ^p+q→Y by means of properties of its sections (Theorem 2).     

One Example in Monotone Approximation

For each non-negative integerna functionf=f_nis constructed such thatfhas a continuous and non-negative derivativef′ onI :=[−1, 1] and[formula]whereE_n(f′) (E⁽¹⁾_n+1(f)) is the value of the best uniform approximation onIof the functionf′ (f) by arbitrary (monotone onI) algebraic polynomials of degree ?n(n+1).     

Approximation by antiderivatives

Let Ω ?C be an open set with simply connected components and suppose that the functionφ is holomorphic on Ω. We prove the existence of a sequence {φ ^(?n)} ofn-fold antiderivatives (i.e., we haveφ ⁽⁰⁾(z)∶=φ(z) andφ ^(?n)(z)=dφ ^(?n?1)(z)/dz for alln ∈ N₀ and z ∈ Ω) such that the following properties hold:

1. For any compact setB ?Ω with connected complement and any functionf that is continuous onB and holomorphic in its interior, there exists a sequence {n _k} such that {φ^?nk} converges tof uniformly onB.
2. For any open setU ?Ω with simply connected components and any functionf that is holomorphic onU, there exists a sequence {m _k} such that {φ^?mk} converges tof compactly onU.
3. For any measurable setE ?Ω and any functionf that is measurable onE, there exists a sequence {p _k} such that {φ ^(-Pk)} converges tof almost everywhere onE.

     

Vertical versus horizontal Poincaré inequalities on the Heisenberg group

Let ? = 〈a, b|a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b〉 be the discrete Heisenberg group, equipped with the left-invariant word metric d _W (·, ·) associated to the generating set {a, b, a ^?1, b ^?1}. Letting B _n = {x ∈ ?: d _W (x, e _?) ? n} denote the corresponding closed ball of radius n ∈ ?, and writing c = [a, b] = aba ^?1 b ^?1, we prove that if (X, ‖ · ‖X) is a Banach space whose modulus of uniform convexity has power type q ∈ [2,∞), then there exists K ∈ (0, ∞) such that every f: ? → X satisfies $$\sum\limits_{k = 1}^{{n^2}} {\sum\limits_{x \in {B_n}} {\frac{{\left\| {f(x{c^k}) - f(x)} \right\|_X^q}}{{{k^{1 + q/2}}}}} } \leqslant K\sum\limits_{x \in {B_{21n}}} {(\left\| {f(xa) - f(x)} \right\|_X^q + \left\| {f(xb) - f(x)} \right\|_X^q)} $$ . It follows that for every n ∈ ? the bi-Lipschitz distortion of every f: B _n → X is at least a constant multiple of (log n)^1/q , an asymptotically optimal estimate as n → ∞.     

Maximum modulus sets in pseudoconvex boundaries

LedD be a strictly pseudoconvex domain in ? ⁿ withC ^∞ boundary. We denote byA ^∞(D) the set of holomorphic functions inD that have aC ^∞ extension to \(\bar D\) . A closed subsetE of ?D is locally a maximum modulus set forA ^∞(D) if for everyp∈E there exists a neighborhoodU ofp andf∈A ^∞(D∩U) such that |f|=1 onE∩U and |f|<1 on \(\bar D \cap U\backslash E\) . A submanifoldM of ?D is an interpolation manifold ifT _p (M)?T _p ^c (?D) for everyp∈M, whereT _p ^c (?D) is the maximal complex subspace of the tangent spaceT _p (?D). We prove that a local maximum modulus set forA ^∞ (D) is locally contained in totally realn-dimensional submanifolds of ?D that admit a unique foliation by (n?1)-dimensional interpolation submanifolds. LetD =D ₁ x ... xD _r ? ? ⁿ whereD _i is a strictly pseudoconvex domain withC ^∞ boundary in ? ⁿ _i ,i=1,…,r. A submanifoldM of ?D ₁×…×?D _r verifies the cone condition if \(II_p (T_p (M)) \cap \bar C[Jn_1 (p),...,Jn_r (p)] = \{ 0\} \) for everyp∈M, wheren _i (p) is the outer normal toD _i atp, J is the complex structure of ? ⁿ , \(\bar C[Jn_1 (p),...,Jn_r (p)]\) is the closed positive cone of the real spaceV _p generated byJ _{n 1}(p),…,J _{n r}(p), and II _p is the orthogonal projection ofT _p (?D) onV _p . We prove that a closed subsetE of ?D ₁×…×?D _r which is locally a maximum modulus set forA ^∞(D) is locally contained inn-dimensional totally real submanifolds of ?D ₁×…×?D _r that admit a foliation by (n?1)-dimensional submanifolds such that each leaf verifies the cone condition at every point ofE. A characterization of the local peak subsets of ?D ₁×…×?D _r is also given.     

Function algebras on which homomorphisms are point evaluations on sequences

In the study of the spectrum of a subalgebraA ofC(X), whereX is a completely regular Hausdorff space, a key question is, whether each homomorphism ?:A→R has the point evaluation property for sequences inA, that is whether, for each sequence (f _n ) inA, there exists a pointa inX such that ?(f _n )=f _n (a) for alln. In this paper it is proved that all algebras, which are closed under composition with functions inC ^∞ (R) and have a certain local property, have the point evaluation property for sequences. Such algebras are, for instance, the spaceC ^m (E) (m=0,1,...,∞) ofC ^m -functions on any real locally convex spaceE. This result yields in a trivial manner that each homomorphism ? onA is a point evaluation, ifX is Lindelöf or ifA contains a sequence which separates points inX. Further, also a well known result as well as some new ones are obtained as a consequence of the main theorem.     

О последовательност ях сумм Фурье-Уолша ограниченных функци й

Let L^∞ denote the space of measurable 1-periodic essentially bounded functionsf(x) with ∥f∥=vrai sup ¦f(x)¦,S _k (f, x) thek-th partial sum of the Walsh-Fourier series off(x),L _k thek-th Lebesgue constant. The following theorem is proved. Theorem. Letλ={λ _K } be a sequence of nonnegative numbers, $$\left\| \lambda \right\|_1 = \mathop \sum \limits_{k = 1}^\infty \lambda _k< \infty ,\left\| \lambda \right\|_2 = (\mathop \sum \limits_{k = 1}^\infty \lambda _k^2 )^{1/2} ,m = log[(\left\| \lambda \right\|_1 /\left\| \lambda \right\|_2 )]$$ .Then for an arbitrary function f∈L ^∞ the following inequalities hold true $$\begin{gathered} \left\| {\mathop \sum \limits_{k = 1}^\infty \lambda _k \left| {S_k (f,x)} \right|} \right\| \leqq \mathop \sum \limits_{k = 1}^\infty \lambda _k (L_{[k2 - 2m]} + c)\left\| f \right\|, \hfill \\ \hfill \\ \mathop \sum \limits_{k = 1}^\infty \lambda _k \left\| {S_k (f)} \right\| \leqq \mathop \sum \limits_{k = 1}^\infty \lambda _k (L_{[k2 - m]} + c)\left\| f \right\| \hfill \\ \end{gathered} $$ , where[y] denotes integral part of a number y>0 and c is an absolute constant. A corollary of the above theorem is that for each functionfεL ^∞ the Lebesgue estimate can be refined for a certain sequence of indices, while the growth order of Lebesgue constants along that sequence can be arbitrarily close to the logarithmic one. “In the mean”, however, the Lebesgue estimate is exact. A further corollary deals with strong summability.     

Multiobjective programming and penalty functions

This paper generalizes the penalty function method of Zang-will for scalar problems to vector problems. The vector penalty function takes the form $$g(x,\lambda ) = f(x) + \lambda ^{ - 1} P(x)e,$$ wheree ?R ^m, with each component equal to unity;f:R ⁿ →R ^m, represents them objective functions {f ⁱ} defined onX \( \subseteq \) R ⁿ; λ ∈R ¹, λ>0;P:R ⁿ →R ¹ X \( \subseteq \) Z \( \subseteq \) R ⁿ,P(x)≦0, ∨x ∈R ⁿ,P(x) = 0 ?x ∈X. The paper studies properties of {E (Z, λ _r )} for a sequence of positive {λ _r } converging to 0 in relationship toE(X), whereE(Z, λ _r ) is the efficient set ofZ with respect tog(·, λ_r) andE(X) is the efficient set ofX with respect tof. It is seen that some of Zangwill's results do not hold for the vector problem. In addition, some new results are given.     

Generalized monotone approximation inL p Space

Letf(x) ∈L _p[0,1], 1?p? ∞. We shall say that functionf(x)∈Δ^k (integerk?1) if for anyh ∈ [0, 1/k] andx ∈ [0,1?kh], we have Δ _h ^k f(x)?0. Denote by ∏ _n the space of algebraic polynomials of degree not exceedingn and define $$E_{n,k} (f)_p : = \mathop {\inf }\limits_{\mathop {P_n \in \prod _n }\limits_{P_n^{(\lambda )} \geqslant 0} } \parallel f(x) - P_n (x)\parallel _{L_p [0,1]} .$$ We prove that for any positive integerk, iff(x) ∈ Δ ^k ∩ L _p[0, 1], 1?p?∞, then we have $$E_{n,k} (f)_p \leqslant C\omega _2 \left( {f,\frac{1}{n}} \right)_p ,$$ whereC is a constant only depending onk.     

О суммируемости обоб щенных рядов Фурье

Consider a functionf satisfying the condition (1) $$\left| x \right|^\alpha f(x) \in L( - \pi ,\pi ),\alpha > 0,$$ , and define the positive integerm by the inequalitiesm ?1<α≦m. The trigonometric series Σ _n=1 ^∞ (a _n cosnx+-b _n sinnx) with coefficients $$\begin{gathered} a_n = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f(t)\left( {\cos nt - \sum\limits_{j = 0}^{[(m - 1)/2]} {\frac{{( - 1)^j (nt)^{2j} }}{{(2j)!}}} } \right)dt,} \hfill \\ b_n = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f(t)\left( {\sin nt - \sum\limits_{j = 1}^{[m/2]} {\frac{{( - 1)^{j + 1} (nt)^{2j - 1} }}{{(2j - 1)!}}} } \right)dt} \hfill \\ \end{gathered} $$ is then called the generalized Fourier series ofmth order off. The following result is proved. Let the 2π-periodic functionf satisfy condition (1) and letт ?1 < α≦m. Then the generalized Fourier series ofmth order off is summable almost everywhere tof(x) by the (C, α)-method. For an arbitrary α∈(0, 1) condition (1) is sharp.     

A Quantitative Version of the Non-Abelian Idempotent Theorem

Suppose that G is a finite group and f is a complex-valued function on G. f induces a (left) convolution operator from L ²(G) to L ²(G) by g ? f *g{g \mapsto f \ast g} where