Multipliers for the Weyl transform and Laguerre expansions

Let P_k denote the projection of L²(R ^R ) onto the kth eigenspace of the operator (-δ+?x?² andS _N ^α =(1/A _N ^α )Σ _k ^N =0A _N?k ^α P _k . We study the multiplier transformT _N ^α for the Weyl transform W defined byW(T _N ^αf )=S _n ^αW(f) . Applications to Laguerre expansions are given.     

A quadratic approximation for protein sequence to structure mapping

The existence and representations of some generalized inverses, includingA _{T, *} ⁽²⁾ ,A _{T, *} ^(1,2) ,A _{T, *} ^(2,3) ,A _*,S ⁽²⁾ ,A _*,S ^(1,2) andA _*,S ^(2,4) , are showed. As applications, the perturbation theory for the generalized inverseA _T,S ⁽²⁾ and the perturbation bound for unique solution of the general restricted systemAx=b (dim (AT)=dimT,b∈AT andx∈T) are studied. Moreover, a characterization and representation of the generalized inverseA _{T, *} ^Emphasis>(2) is obtained.     

Comparison Inequalities for One Sided Normal Probabilities

Several sharp upper and lower bounds for the ratio of two normal probabilities $\mathbb{P}\Biggl(\,\bigcap_{i=1}^{n}\bigl\{\xi^{(1)}_i\leq \mu_i\bigr\}\Biggr)\Big/\mathbb{P}\Biggl(\,\bigcap_{i=1}^{n}\bigl\{\xi^{(0)}_i\leq \mu_i\bigr\}\Biggr)$ are given in this paper for various cases, where (ξ ₁ ⁽⁰⁾ ,ξ ₂ ⁽⁰⁾ ,…,ξ _n ⁽⁰⁾ ) and (ξ ₁ ⁽¹⁾ ,ξ ₂ ⁽¹⁾ , …,ξ _n ⁽¹⁾ ) are standard normal random variables with covariance matrices R ⁰=(r _ij ⁰ ) and R ¹=(r _ij ¹ ), respectively.     

On the chromatic numbers of spheres in ℝ n

Let χ(S _r ^n?1 )) be the minimum number of colours needed to colour the points of a sphere S _r ^n?1 of radius $r \geqslant \tfrac{1} {2}$ in ? ⁿ so that any two points at the distance 1 apart receive different colours. In 1981 P. Erd?s conjectured that χ(S _r ^n?1 )→∞ for all $r \geqslant \tfrac{1} {2}$ . This conjecture was proved in 1983 by L. Lovász who showed in [11] that χ(S _r ^n?1 ) ≥ n. In the same paper, Lovász claimed that if $r < \sqrt {\frac{n} {{2n + 2}}}$ , then χ(S _r ^n?1 ) ≤ n+1, and he conjectured that χ(S _r ^n?1 ) grows exponentially, provided $r \geqslant \sqrt {\frac{n} {{2n + 2}}}$ . In this paper, we show that Lovász’ claim is wrong and his conjecture is true: actually we prove that the quantity χ(S _r ^n?1 ) grows exponentially for any $r > \tfrac{1} {2}$ .     

On positive linear interpolation operators

В этой работе мы даем о бобщение понятия нор мальной системы точек, введен ного Фейером [3]. Наше определ ение включает и случа й бесконечного интерв ала (0, ∞). Доказано, в частности, что систе ма точек 0<x ₁ ⁽ⁿ⁾ /(n)<... _n ⁽ⁿ⁾ <∞ является нормальной в смысле нашего определения тогда и т олько тогда, когда вып олняются оценки — фиксированное чис ло, 0≦?<1. Мы доказываем, что есл и точкиx _k ⁽ⁿ⁾ /(n) являются ну лями многочлена ЛагерраL _n ^(α) (x), то они образуют норма льную систему в том и т олько том случае, когда ?1<α≦0. Мы получаем, таким обр азом, положительный интерполяционный пр оцесс для каждой нормальной системы т очек и устанавливаем теорему сходимости для того с лучая, когда эти точки являются ну лямиL _n ^(α) (x) при — 1相似文献   

Convolutions of random measures on compact groups

LetG be a compact group andM ₁(G) be the convolution semigroup of all Borel probability measures onG with the weak topology. We consider a stationary sequence {μ _n } _n=?∞ ^+∞ of random measures μ _n =μ _n (ω) inM ₁(G) and the convolutions $$v_{m,n} (\omega ) = \mu _m (\omega )* \cdots *\mu _{n - 1} (\omega ), m< n$$ and $$\alpha _n^{( + k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n,n + i} (\omega ),} \alpha _n^{( - k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n - i,n} (\omega )} $$ We describe the setsA _m ⁺ (ω) andA _n ⁺ (ω) of all limit points ofv _m,n(ω) asm→?∞ orn→+∞ and the setA _∞(ω) of its two-sided limit points for typical realizations of {μ _n (ω)} _n=?∞ ^+∞ . Using an appropriate random ergodic theorem we study the limit random measures ρ _n ^(±) (ω)=lim _k→∞ α _n ^(±k) (ω).     

On the eigenvalues of the Sturm-Liouville operator with potentials from Sobolev spaces

We study the asymptotic behavior of the eigenvalues the Sturm-Liouville operator Ly = ?y″ + q(x)y with potentials from the Sobolev space W ₂ ^θ?1 , θ ≥ 0, including the nonclassical case θ ∈ [0, 1) in which the potential is a distribution. The results are obtained in new terms. Let s _2k (q) = λ _k ^1/2 (q) ? k, s _2k?1(q) = μ _k ^1/2 (q) ? k ? 1/2, where {λ _k } ₁ ^∞ and {μ _k } ₁ ^∞ are the sequences of eigenvalues of the operator L generated by the Dirichlet and Dirichlet-Neumann boundary conditions, respectively,. We construct special Hilbert spaces t ₂ ^θ such that the mapping F:W ₂ ^θ?1 →t ₂ ^θ defined by the equality F(q) = {s _n } ₁ ^∞ is well defined for all θ ≥ 0. The main result is as follows: for θ > 0, the mapping F is weakly nonlinear, i.e., can be expressed as F(q) = Uq + Φ(q), where U is the isomorphism of the spaces W ₂ ^θ?1 and t ₂ ^θ , and Φ(q) is a compact mapping. Moreover, we prove the estimate ∥Ф(q)∥_τ ≤ C∥q∥_θ?1, where the exact value of τ = τ(θ) > θ ? 1 is given and the constant C depends only on the radius of the ball ∥q∥_θ? ≤ R, but is independent of the function q varying in this ball.     

Limit ultraspherical series and their approximation properties

We study new series of the form $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ in which the general term $f_k^{ - 1} \hat P_k^{ - 1} (x)$ , k = 0, 1, …, is obtained by passing to the limit as α→?1 from the general term $\hat f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)$ of the Fourier series $\sum\nolimits_{k = 0}^\infty {f_k^\alpha \hat P_k^{\alpha ,\alpha } (x)} $ in Jacobi ultraspherical polynomials $\hat P_k^{\alpha ,\alpha } (x)$ generating, for α> ?1, an orthonormal system with weight (1 ? x ₂)α on [?1, 1]. We study the properties of the partial sums $S_n^{ - 1} (f,x) = \sum\nolimits_{k = 0}^n {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ of the limit ultraspherical series $\sum\nolimits_{k = 0}^\infty {f_k^{ - 1} \hat P_k^{ - 1} (x)} $ . In particular, it is shown that the operator S _n ^?1 (f) = S _n ^?1 (f, x) is the projection onto the subspace of algebraic polynomials p _n = p _n (x) of degree at most n, i.e., S _n (p _n ) = p _n ; in addition, S _n ^?1 (f, x) coincides with f(x) at the endpoints ±1, i.e., S _n ^?1 (f,±1) = f(±1). It is proved that the Lebesgue function Λ _n (x) of the partial sums S _n ^?1 (f, x) is of the order of growth equal to O(ln n), and, more precisely, it is proved that $\Lambda _n (x) \leqslant c(1 + \ln (1 + n\sqrt {1 - x^2 } )), - 1 \leqslant x \leqslant 1$ .     

Determinants of parabolic bundles on Riemann surfaces

LetX be a compact Riemann surface andM _s ^p (X) the moduli space of stable parabolic vector bundles with fixed rank, degree, rational weights and multiplicities. There is a natural Kähler metric onM _s ^p (X). We obtain a natural metrized holomorphic line bundle onM _s ^p (X) whose Chern form equalsmr times the Kähler form, wherem is the common denominator of the weights andr the rank.     

ON possible Turán densities

The Turán density π(F) of a family F of k-graphs is the limit as n → ∞ of the maximum edge density of an F-free k-graph on n vertices. Let Π _∞ ^(k) consist of all possible Turán densities and let Π _fin ^(k) ? Π _∞ ^(k) be the set of Turán densities of finite k-graph families. Here we prove that Π _fin ^(k) contains every density obtained from an arbitrary finite construction by optimally blowing it up and using recursion inside the specified set of parts. As an application, we show that Π _fin ^(k) contains an irrational number for each k ≥ 3. Also, we show that Π _∞ ^(k) has cardinality of the continuum. In particular, Π _∞ ^(k) ≠ Π _fin ^(k) .     

An example of asymmetry of convolution operators

Esistono un gruppo compatto non commutativoG ^～ ed un operatore di convoluzioneT tale che: perp∈[2,4] e perq∈[1,2),T∈L _p ^p (G ^～) eT?L _q ^q (G ^～).     

An abstract functional differential equation and a related nonlinear Volterra equation

We study the existence, uniqueness, regularity and dependence upon data of solutions of the abstract functional differential equation 1 $$\frac{{du}}{{dt}} + Au \ni G(u) (0 \leqq t \leqq T), u(0) = x,$$ , whereT>0 is arbitrary,A is a givenm-accretive operator in a real Banach spaceX, and \(G:C([0,T]; \overline {D(A)} ) \to L^1 (0, T; X)\) is a given mapping. This study provides simple proofs of generalizations of results by several authors concerning the nonlinear Volterra equation 2 $$u(t) + b * Au(t) \ni F(t) (0 \leqq t \leqq T),$$ , for the case in which X is a real Hilbert space. In (2) the kernelb is real, absolutely continuous on [0,T],b*g(t)=∫ ₀ ¹ (t?s)g(s)ds, andf∈W ^1,1(0,T;X).     

The index of singular integral operators in reflexive Orlicz spaces

We consider the Banach algebra $\mathfrak{A}$ of singular integral operators with matrix piecewise continuous coefficients in the reflexive Orlicz spaceL _M ⁿ (Γ). We assume that Γ belongs to a certain wide subclass of the class of Carleson curves; this subclass includes curves with cusps, as well as curves of the logarithmic spiral type. We obtain an index formula for an arbitrary operator from the algebra $L_M^n (\Gamma )$ in terms of the symbol of this operator.     

Spectral asymptotics and trace formulas for differential operators with unbounded coefficients

In the space L ²[0, π], we consider the operators $$ L = L_0 + V, L_0 = - y'' + (\nu ^2 - 1/4)r^{ - 2} y (\nu \geqslant 1/2) $$ with the Dirichlet boundary conditions. The potential V is the operator of multiplication by a function (in general, complex-valued) in L ²[0, π] satisfying the condition $$ \int\limits_0^\pi {r^\varepsilon } (\pi - r)^\varepsilon |V(r)|dr < \infty , \varepsilon \in [0,1] $$ . We prove the trace formula Σ _n=1 ^∞ [µ _n ? λ _n ? Σ _k=1 ^m α _k ⁽ⁿ⁾ ] = 0.     

On quasinormal,subnormal, and hyponormal Toeplitz operators

Let ? be a non-constant function inL ^∞(D) such thatφ=φ ₁+φ ₂, whereφ ₁ is an element in the Bergman spaceL _a ² (D), and \(\phi _2 \in \overline {L_a^2 (D)} \) , the space of all complex conjugates of functions inL _a ² (D). In this paper, it is shown that if 1 is an element in the closure of the range of the self-commutator ofT _?, \(T_{\bar \phi } T_\phi - T_\phi T\phi \) , then the Toeplitz operatorT _? defined ofL _a ² (D) is not quasinormal. Moreover, if \(\phi = \psi + \lambda \bar \psi \) , whereψ∈ H ^∞(D), and λεC, it is proved that ifT _? is quasinormal, thenT _? is normal. Also, the spectrum of a class of pure hyponormal Toeplitz operators is determined.     

Random and algebraic permutations’ statistics

An algebraic permutation $\hat{A}\in S(N=n^{m})$ is the permutation of the N points of the finite torus ? _n ^m , realized by a linear operator A∈SL(m,? _n ). The statistical properties of algebraic permutations are quite different from those of random permutations of N points. For instance, the period length T(A) grows superexponentially with N for some (random) permutations A of N elements, whereas $T(\hat{A})$ is bounded by a power of N for algebraic permutations  $\hat{A}$ . The paper also contains a strange mean asymptotics formula for the number of points of the finite projective line P¹(? _n ) in terms of the zeta function.     

On coefficients of null-series and on sets of uniqueness of trigonometric and Walsh systems

В работе доказываютс я следующие утвержде ния. Теорема I.Пусть ? _n ↓0u \(\sum\limits_{n = 0}^\infty {\varepsilon _n^2 = + \infty } \) .Тогд а существует множест во Е?[0, 1]с μЕ=0 такое что:1. Существует ряд \(\sum\limits_{n = 0}^\infty {a_n W_n } (t)\) с к оеффициентами ¦а _n ¦≦{in¦n¦, который сх одится к нулю всюду вне E и ε∥a_n∥>0.2. Если b _n ¦=о(ε _n )и ряд \(\sum\limits_{n = 0}^\infty {b_n W_n (t)} \) сх одится к нулю всюду вн е E за исключением быть может некоторого сче тного множества точе к, то b _n =0для всех п. Теорема 3.Пусть ? _n ↓0u \(\mathop {\lim \sup }\limits_{n \to \infty } \frac{{\varepsilon _n }}{{\varepsilon _{2n} }}< \sqrt 2 \) Тогд а существует множест во E?[0, 1] с υ E=0 такое, что:

1. Существует ряд \(\sum\limits_{n = - \infty }^{ + \infty } {a_n e^{inx} ,} \sum\limits_{n = - \infty }^{ + \infty } {\left| {a_n } \right|} > 0,\) кот орый сходится к нулю в сюду вне E и ¦a_n≦¦n¦ для n=±1, ±2, ...
2. Если ряд \(\sum\limits_{n = - \infty }^{ + \infty } {b_n e^{inx} } \) сходится к нулю всюду вне E и ¦b_v¦=о(ε ¦n¦), то b_n=0 для всех я. Теорема 5. Пусть послед овательности S⁽¹⁾={ε ₀ ⁽¹⁾ , ε ₁ ⁽¹⁾ , ε ₂ ⁽¹⁾ , ...} u S²=ε ₀ ⁽²⁾ , ε ₁ ⁽²⁾ . ε ₂ ⁽²⁾ монотонно стремятся к нулю, \(\mathop {\lim \sup }\limits_{n \to \infty } \varepsilon ^{(i)} /\varepsilon _{2n}^{(i)}< 2,i = 1,2\) , причем \(\mathop {\lim }\limits_{n \to \infty } \varepsilon _n^{(2)} /\varepsilon _n^{(i)} = + \infty \) . Тогда для каждого ε>O н айдется множество Е? [-π,π], μE >2π — ε, которое является U(S¹), но не U(S¹) — множеством для тригонометричес кой системы. Аналог теоремы 5 для си стемы Уолша был устан овлен в [7].

     

On the relation of generalized Valiron summability to Cesàro summability

A family (V _a ^k ) of summability methods, called generalized Valiron summability, is defined. The well-known summability methods (B_α,γ), (E _ρ, (T_α), (S _β) and (V _a) are members of this family. In §3 some properties of the (B _α,γ) and (V _a ^k ) transforms are established. Following Satz II of Faulhaber (1956) it is proved that the members of the (V _a ^k ) family are all equivalent for sequences of finite order. This paper is a good illustration of the use of generalized Boral summability. The following theorem is established: Theorem.If s _n (n ≥ 0) isa real sequence satisfying $$\mathop {lim}\limits_{ \in \to 0 + } \mathop {lim inf}\limits_{m \to \infty } \mathop {min}\limits_{m \leqslant n \leqslant m \in \sqrt m } \left( {\frac{{S_n - S_m }}{{m^p }}} \right) \geqslant 0(\rho \geqslant 0)$$ , and if s_n →s (V _a ^k ) thens _n → s (C, 2ρ).     

On the Brauer group of the product of a torus and a semisimple algebraic group

Let T be a torus (not assumed to be split) over a field F, and denote by nH _et ² (X,{ie375-1}) the subgroup of elements of the exponent dividing n in the cohomological Brauer group of a scheme X over the field F. We provide conditions on X and n for which the pull-back homomorphism nH _et ² (T,{ie375-2}) → _n H _et ² (X × _F T, {ie375-3}) is an isomorphism. We apply this to compute the Brauer group of some reductive groups and of non-singular affine quadrics. Apart from this, we investigate the p-torsion of the Azumaya algebra defined Brauer group of a regular affine scheme over a field F of characteristic p > 0.     

Weak and strong convergence to common fixed points of non-self nonexpansive mappings

SupposeK is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach spaceE withP as a nonexpansive retraction. LetT ₁,T ₂ andT ₃:K → E be nonexpansive mappings with nonempty common fixed points set. Letα _n ,β _n ,γ _n ,α _n ^′ ,β _n ^′ ,γ _n ^′ ,α _n ^′′ ,β _n ^′′ andγ _n ^′′ be real sequences in [0, 1] such thatα _n +β _n +γ _n =α _n ^′ +β _n ^′ +γ _n ^′ =α _n ^′′ +β _n ^′′ +γ _n ^′′ = 1, starting from arbitraryx ₁ ∈ K, define the sequencex _n by $$\left\{ \begin{gathered} z_n = P(\alpha ''_n T_1 x_n + \beta ''_n x_n + \gamma ''_n w_n ) \hfill \\ y_n = P(\alpha _n^\prime T_2 z_n + \beta _n^\prime x_n + \gamma _n^\prime v_n ) \hfill \\ x_{n + 1} = P(\alpha _n T_3 y_n + \beta _n x_n + \gamma _n u_n ) \hfill \\ \end{gathered} \right.$$ with the restrictions $\sum\limits_{n = 1}^\infty {\gamma _n }< \infty , \sum\limits_{n = 1}^\infty \gamma _n^\prime< \infty ,\sum\limits_{n = 1}^\infty {\gamma ''_n }< \infty $ . (i) If the dual E* ofE has the Kadec-Klee property, then weak convergence of ax _n to somex* ∈ F(T ₁) ∩F(T ₂) ∩ (T ₃) is proved; (ii) IfT ₁,T₂ andT ₃ satisfy condition (A′), then strong convergence ofx _n to some x* ∈F(T ₁) ∩F(T ₂) ∩ (T ₃) is obtained.