Ultrapowers ofL 1(μ) and the subsequence splitting principle

We study the ultrapowers $L_1 (\mu )_\mathfrak{U} $ of aL ₁(μ) space, by describing the components of the well-known representation $L_1 (\mu )_\mathfrak{U} = L_1 (\mu _\mathfrak{U} ) \oplus _1 L_1 (\nu _\mathfrak{U} )$ , and we give a representation of the projection from $L_1 (\mu )_\mathfrak{U} $ onto $L_1 (\mu _\mathfrak{U} )$ . Moreover, the subsequence splitting principle forL ₁(μ) motivates the following question: if $\mathfrak{V}$ is an ultrafilter on ? and $[f_i ] \in L_1 (\mu )_\mathfrak{V} $ , is it possible to find a weakly convergent sequence (g _i ) ?L ₁(μ) following $\mathfrak{V}$ and a disjoint sequence (h _i ) ?L ₁(μ) such that [f _i ]=[g _i ]+[h _i ]? If $\mathfrak{V}$ is a selective ultrafilter, we find a positive answer by showing that $f = [f_i ] \in L_1 (\mu )_\mathfrak{V} $ belongs to $L_1 (\mu _{_\mathfrak{V} } )$ if and only if its representatives {f _i } are weakly convergent following $\mathfrak{V}$ and $f \in L_1 (\nu _\mathfrak{V} )$ if and only if it admits a representative consisting of pairwise disjoint functions. As a consequence, we obtain a new proof of the subsequence splitting principle. If $\mathfrak{V}$ is not a p-point then the above characterizations of $L_1 (\nu _{_\mathfrak{V} } )$ and $L_1 (\nu _{_\mathfrak{V} } )$ fail and the answer to the question is negative.     

The index of centralizers of elements in classical Lie algebras

The index of a finite-dimensional Lie algebra $\mathfrak{g}$ is the minimum of dimensions of the stabilizers $\mathfrak{g}_\alpha $ over all covectors $\alpha \in \mathfrak{g}^ * $ . Let $\mathfrak{g}$ be a reductive Lie algebra over a field $\mathbb{K}$ of characteristic ≠ = 2. Élashvili conjectured that the index of $\mathfrak{g}_\alpha $ is always equal to the index, or, which is the same, the rank of $\mathfrak{g}$ . In this article, Élashvili’s conjecture is proved for classical Lie algebras. Furthermore, it is shown that if $\mathfrak{g} = \mathfrak{g}\mathfrak{l}_n $ or $\mathfrak{g} = \mathfrak{s}\mathfrak{p}_{2n} $ and $e \in \mathfrak{g}$ is a nilpotent element, then the coadjoint action of $\mathfrak{g}_e $ has a generic stabilizer. For $\mathfrak{g}$ , we give examples of nilpotent elements $e \in \mathfrak{g}$ such that the coadjoint action of $\mathfrak{g}_e $ does not have a generic stabilizer.     

The argument shift method and the Gaudin model

We construct a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra U ( ) of a semisimple Lie algebra . This family is parameterized by finite sequences μ, z ₁, ..., z _n , where μ ∈ ^* and z _i ∈ ℂ. The construction presented here generalizes the famous construction of the higher Gaudin Hamiltonians due to Feigin, Frenkel, and Reshetikhin. For n = 1, the corresponding commutative subalgebras in the Poisson algebra S( ) were obtained by Mishchenko and Fomenko with the help of the argument shift method. For commutative algebras of our family, we establish a connection between their representations in the tensor products of finite-dimensional -modules and the Gaudin model. __________ Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 40, No. 3, pp. 30–43, 2006 Original Russian Text Copyright ? by L. G. Rybnikov     

Embedded Spaces of Trigonometric Splines and Their Wavelet Expansion

With each infinite grid X: ? < x _?1 < x ₀ < x ₁ < ? we associate the system of trigonometric splines $\{ \mathfrak{T}_j^B \}$ of class C ¹(α, β), the linear space $$T^B (X)\mathop = \limits^{def} \{ \tilde u|\tilde u = \sum\limits_j {c_j \mathfrak{T}_j^B } \quad \forall c_j \in \mathbb{R}^1 \} ,$$ and the functionals g ⁽ⁱ⁾ ∈ (C ¹(α, β))* with the biorthogonality property: $\left\langle {g(i),\mathfrak{T}_j^B } \right\rangle = \delta _{i,j}$ (here $\alpha \mathop = \limits^{def} \lim _{j \to - \infty } x_j ,\quad \beta \mathop = \limits^{def} \lim _{j \to + \infty } x_j$ ). For nested grids $\bar X \subset X$ , we show that the corresponding spaces $T^B (\bar X)$ are embedded in $T^B (X)$ and obtain decomposition and reconstruction formulas for the spline-wavelet expansion $T^B (X) = T^B (\bar X)\dot + W$ derived with the help of the system of functionals indicated above.     

A Characterization of Projective Spaces by a Set of Planes

This note deals with the following question: How many planes of a linear space (P, $\mathfrak{L}$ ) must be known as projective planes to ensure that (P, $\mathfrak{L}$ ) is a projective space? The following answer is given: If for any subset M of a linear space (P, $\mathfrak{L}$ ) the restriction (M, $\mathfrak{L}$ )(M)) is locally complete, and if for every plane E of (M, $\mathfrak{L}$ (M)) the plane $\bar E$ generated by E is a projective plane, then (P, $\mathfrak{L}$ ) is a projective space (cf. 5.6). Or more generally: If for any subset M of P the restriction (M, $\mathfrak{L}$ (M)) is locally complete, and if for any two distinct coplanar lines G1, G2 ∈ $\mathfrak{L}$ (M) the lines $\bar G_1 ,\bar G_2 \varepsilon \mathfrak{L}$ generated by G1, G2 have a nonempty intersection and $\overline {G_1 \cup {\text{ }}G_2 }$ satisfies the exchange condition, then (P, $\mathfrak{L}$ ) is a generalized projective space.     

Approximation by M. Riesz Kernels in L p for p<1

Let α > 0. We consider the linear span $\mathfrak{X}_\alpha \left( {\mathbb{R}^n } \right)$ of scalar Riesz's kernels $\left\{ {\tfrac{1}{{\left| {x - a} \right|^\alpha }}} \right\}_{a \in \mathbb{R}^n }$ and the linear span $\mathfrak{Y}_\alpha \left( {\mathbb{R}^n } \right)$ of vector Riesz's kernels $\left\{ {\tfrac{1}{{\left| {x - a} \right|^{\alpha + 1} }}\left( {x - a} \right)} \right\}_{a \in \mathbb{R}^n }$ . We study the following problems. (1) When is the intersection $\mathfrak{X}_\alpha \left( {\mathbb{R}^n } \right) \cap L^p \left( {\mathbb{R}^n } \right)$ dense in L^p(?ⁿ)? (2) When is the intersection $\mathfrak{Y}_\alpha \left( {\mathbb{R}^n } \right) \cap L^p \left( {\mathbb{R}^n ,\mathbb{R}^n } \right)$ dense in L^p(?ⁿ, ?ⁿ)? Bibliography: 15 titles.     

The periodic groups saturated by finitely many finite simple groups

Denote by $\mathfrak{M}$ the set whose elements are the simple 3-dimensional unitary groups U ₃(q) and the linear groups L ₃(q) over finite fields. We prove that every periodic group, saturated by the groups of a finite subset of $\mathfrak{M}$ , is finite.     

A Criterion for Weak Generalized Localization in the Class L_1 for Multiple Trigonometric Series from the Viewpoint of Isometric Transformations

In this paper, we study the problem of the variation (if any) of the sets of convergence and divergence everywhere or almost everywhere of a multiple Fourier series (integral) of a function $f \in L_p $ , $p \geqslant 1$ , $f(x) = 0$ , on a set of positive measure $\mathfrak{A} \subset \mathbb{T}^N = [ - \pi ,\pi )^N $ , $N \geqslant 2$ , depending on the rotation of the coordinate system, i.e., depending on the element $\tau \in \mathcal{F}$ , where $\mathcal{F}$ is the rotation group about the origin in $\mathbb{R}^N $ . This problem has been reduced to the study of the change in the geometry of the sets $\tau ^{ - 1} (\mathfrak{A}) \cap \mathbb{T}^N $ (where $\tau ^{ - 1} \in \mathcal{F}$ satisfies $\tau ^{ - 1} \cdot \tau = 1$ ) and $\mathbb{T}^N \backslash {\text{supp}}(f \circ \tau )$ depending on the “rotation,” i.e., on $\tau \in \mathcal{F}$ . In the present paper, we consider two settings of this problem (depending on the sense in which the Fourier series of the function $f \circ \tau $ is understood) and give (for both cases) possible solutions of the problem in the class $L_1 (\mathbb{T}^N )$ , $N \geqslant 2$ .     

ОБ ИНтЕРпОлИРОВАНИИ В кОМплЕксНОИ ОБлАст И

LetD be a simply connected domain, the boundary of which is a closed Jordan curveγ; \(\mathfrak{M} = \left\{ {z_{k, n} } \right\}\) , 0≦k≦n; n=1, 2, 3, ..., a matrix of interpolation knots, \(\mathfrak{M} \subset \Gamma ; A_c \left( {\bar D} \right)\) the space of the functions that are analytic inD and continuous on \(\bar D; \left\{ {L_n \left( {\mathfrak{M}; f, z} \right)} \right\}\) the sequence of the Lagrange interpolation polynomials. We say that a matrix \(\mathfrak{M}\) satisfies condition (B _m ), \(\mathfrak{M}\) ∈(B _m ), if for some positive integerm there exist a setB _m containingm points and a sequencen _{p p=1} ^∞ of integers such that the series \(\mathop \Sigma \limits_{p = 1}^\infty \frac{1}{{n_p }}\) diverges and for all pairsn _i ,n _j ∈{n _p } _p=1 ^∞ the set \(\left( {\bigcap\limits_{k = 0}^{n_i } {z_{k, n_i } } } \right)\bigcap {\left( {\bigcup\limits_{k = 0}^{n_j } {z_{k, n_j } } } \right)} \) is contained inB _m . The main result reads as follows. {Let D=z: ¦z¦ \(\Gamma = \partial \bar D\) and let the matrix \(\mathfrak{M} \subset \Gamma \) satisfy condition (B_m). Then there exists a function \(f \in A_c \left( {\bar D} \right)\) such that the relation $$\mathop {\lim \sup }\limits_{n \to \infty } \left| {L_n \left( {\mathfrak{M}, f, z} \right)} \right| = \infty $$ holds almost everywhere on γ.     

Restrictions of generalized Verma modules to symmetric pairs

We initiate a new line of investigation on branching problems for generalized Verma modules with respect to reductive symmetric pairs $ \left( {\mathfrak{g},\mathfrak{g}'} \right) $ . In general, Verma modules may not contain any simple module when restricted to a reductive subalgebra. In this article we give a necessary and sufficient condition on the triple $ \left( {\mathfrak{g},\mathfrak{g}',\mathfrak{p}} \right) $ such that the restriction $ {\left. X \right|_{\mathfrak{g}'}} $ always contains simple $ \mathfrak{g}' $ -modules for any $ \mathfrak{g} $ -module X lying in the parabolic BGG category $ {\mathcal{O}^\mathfrak{p}} $ attached to a parabolic subalgebra $ \mathfrak{p} $ of $ \mathfrak{g} $ . Formulas are derived for the Gelfand?CKirillov dimension of any simple module occurring in a simple generalized Verma module. We then prove that the restriction $ {\left. X \right|_{\mathfrak{g}'}} $ is generically multiplicity-free for any $ \mathfrak{p} $ and any $ X \in {\mathcal{O}^\mathfrak{p}} $ if and only if $ \left( {\mathfrak{g},\mathfrak{g}'} \right) $ is isomorphic to (A _n , A _n-1), (B _n , D _n ), or (D _n+1, B _n ). Explicit branching laws are also presented.     

Analytische Fortsetzbarkeit von zufälligen Potenzreihen bezüglich abhängiger Zufallsvariabler

Рассматриваются слу чайная величина \(\mathfrak{X} = (X_n (\omega ))\) , удовлетворяющая усл овиюE(X _n ⁴ )≦M, и соответствующ ий случайный степенн ой ряд \(f_x (z;\omega ) = \mathop \sum \limits_{n = 0}^\infty a_n X_n (\omega )z^n\) . Устанавливаются тео ремы непродолжимост и почти наверное:

1. дляf _x при условиях с лабой мультипликати вности на \(\mathfrak{X}\) ,
2. для \(f_{\tilde x}\) , где \(\mathop \mathfrak{X}\limits^ \sim = (\mathop X\limits^ \sim _n )\) есть подп оследовательность в \(\mathfrak{X}\) ,
3. для по крайней мере од ного из рядовf _x′ илиf _x″ , где \(\mathfrak{X}'\) и \(\mathfrak{X}''\) — некоторые п ерестановки \(\mathfrak{X}\) , выбираемые универс ально, т. е. независимо от коэффициентовa _n .

     

Varieties Generated by Wreath Products of Abelian Groups

This paper is a survey of our recent results concerning metabelian varieties, and more specifically, varieties generated by wreath products of Abelian groups. We give a full classification of cases where sets of wreath products of Abelian groups $ \mathfrak{X} $ Wr $ \mathfrak{Y} $ = { X Wr Y | X ∈ $ \mathfrak{X} $ , Y ∈ $ \mathfrak{Y} $ } and $ \mathfrak{X} $ wr $ \mathfrak{Y} $ = {X wr Y | X ∈ $ \mathfrak{X} $ , Y ∈ $ \mathfrak{Y} $ } generate the product variety $ \mathfrak{X} $ var ( $ \mathfrak{Y} $ ).     

Interpolation of states by vector states on certain operator algebras

LetT be an operator on an infinite dimensional Hilbert space \(\mathcal{H}\) with eigenvectorsv _i , ‖v _i ‖=1,i=1, 2, ..., andsp{v _i ?i∈n} dense in \(\mathcal{H}\) . Suppose that {v _i } is a Schauder basis for \(\mathcal{H}\) . We denote byA _T the ultraweakly closed algebra generated byT andI, the identity operator on \(\mathcal{H}\) . For any nonnegative sequence of scalars \(\left\{ {\alpha ,with = \sum\nolimits_1^\infty {\alpha _1 } = 1} \right\},\) , we associate an ultraweakly (normal) continuous linear functional \(\phi _\alpha = \sum\nolimits_1^\infty {\alpha _j } \omega _v\) where \(\phi _\alpha \left( A \right) = \lim _n \sum\nolimits_1^n {\alpha _j } \omega _v ,\) , and \(\omega _v ,\left( A \right) =< Av_1 ,v_1 >\) for allA∈A _T . We denote the set of all such linear functionals onA TbyF(T). The question that we investigate in this paper is whether each linear functional φ_α inF(T) is a vector state, i.e. does φ_α=ωx for some unit vectorx in \(\mathcal{H}\) ?     

\mathfrak{E} of topological spaces II

In this paper, we obtain analogues, in the situation of \(\mathfrak{E}\) -extensions, of Magill's theorem on lattices of compactifications. We define an epireflective subcategory of the categoryT 2 of all Hausdorff spaces to be admissive (respectively finitely admissive) if for any \(\mathfrak{E}\) -regular spaceX, every Hausdorff quotient of \(\beta _\mathfrak{E} X\) which is Urysohn on \(\beta _\mathfrak{E} X - X\) (respectively which is finitary on \(\beta _\mathfrak{E} X - X\) ) and which is identity onX, has \(\mathfrak{E}\) . We notice that there are many proper epireflective subcategories ofT 2 containing all compact spaces and which are admissive; there are many such which are not admissive but finitely admissive. We prove that when \(\mathfrak{E}\) is a finitely admissive epireflective subcategory ofT 2, then the lattices of finitary \(\mathfrak{E}\) -extensions of two spacesX andY are isomorphic if and only if \(\beta _\mathfrak{E} X - X\) and \(\beta _\mathfrak{E} Y - Y\) are homeomorphic. Further if \(\mathfrak{E}\) is admissive, then the lattices of Urysohn \(\mathfrak{E}\) -extensions ofX andY are isomorphic if and only if \(\beta _\mathfrak{E} X - X\) and \(\beta _\mathfrak{E} Y - Y\) are homeomorphic.     

A-Systems, Independent Functions, and Sets Bounded in Spaces of Measurable Functions

Let $U \subset L_o ([0,1],\mathcal{M},m)$ be a set of Lebesgue measurable functions. Suppose also that two seminormed spaces of real number sequences are given: $\mathcal{A}$ and $\mathcal{B}$ . We study $\left( {\mathcal{A},\mathcal{B}} \right)$ -sets U defined by the classes $\mathcal{A}$ and $\mathcal{B}$ as follows: $\forall a = (a_n ) \in \mathcal{A}, \forall (f_n (t)) \in u^\mathbb{N} $ (or for sequences similar to $(f_n (t))$ ) $\exists E = E(a) \subset [0,1], mE = 1$ such that $\{ a_n f_n (t)\} 1_E (t)\} \in \mathcal{B}, t \in [0,1]$ . We consider three versions of the definition of $\left( {\mathcal{A},\mathcal{B}} \right)$ -sets, one of which is based on functions independent in the probability sense. The case ${\mathcal{B}}=l_\infty$ is studied in detail. It is shown that $({\mathcal{A}},l_\infty)$ -independent sets are sets bounded or order bounded in some well-known function spaces (L _p , L _p,q , etc.) constructed with respect to the Lebesgue measure. A characterization of such sets in terms of seminormed spaces of number sequences is given. The (l ₁,c _°)- and $(\mathcal{A},l_1 )$ -sets were studied by E. M. Nikishin.     

Nikodym Maximal Functions Associated with Variable Planes in {\mathbb{R}^{3}}

Given a vector field ${\mathfrak{a}}$ on ${\mathbb{R}^3}$ , we consider a mapping ${x\mapsto \Pi_{\mathfrak{a}}(x)}$ that assigns to each ${x\in\mathbb{R}^3}$ , a plane ${\Pi_{\mathfrak{a}}(x)}$ containing x, whose normal vector is ${\mathfrak{a}(x)}$ . Associated with this mapping, we define a maximal operator ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^1_{loc}(\mathbb{R}^3)}$ for each ${N\gg 1}$ by $$\mathcal{M}^{\mathfrak{a}}_Nf(x)=\sup_{x\in\tau} \frac{1}{|\tau|} \int_{\tau}|f(y)|\,dy$$ where the supremum is taken over all 1/N ×? 1/N?× 1 tubes τ whose axis is embedded in the plane ${\Pi_\mathfrak{a}(x)}$ . We study the behavior of ${\mathcal{M}^{\mathfrak{a}}_N}$ according to various vector fields ${\mathfrak{a}}$ . In particular, we classify the operator norms of ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^2(\mathbb{R}^3)}$ when ${\mathfrak{a}(x)}$ is the linear function of the form (a ₁₁ x ₁?+?a ₂₁ x ₂, a ₁₂ x ₁?+?a ₂₂ x ₂, 1). The operator norm of ${\mathcal{M}^\mathfrak{a}_N}$ on ${L^2(\mathbb{R}^3)}$ is related with the number given by $$D=(a_{12}+a_{21})^2-4a_{11}a_{22}.$$      

On Uniform Approximation in R2 of Continuous Functions of Two Variables with Bounded Variation in the Sense of Hardy

Introduce the notation: $\mathbb{Z}$ is the set of integers, $\bar {\mathbb{Z}}={\mathbb{Z}} \cup \{-\infty, +\infty\},{\mathbb{R}}_+^2 =\{x=(x_1,x_2) \in {\mathbb{R}}^2; x_1>0,x_2>0\}$ , $g_{k,m} (x,\alpha,h)= \int\limits_0^1 {g_1 (\frac{(k+u)h_1 - x_1}{\alpha_1})g_2(\frac{(m+u)h_2 - x_2}{\alpha_2})}du$ , where $g_i :\mathbb{R} \to \mathbb{R},x \in \mathbb{R}^2 ,\alpha ,h \in \mathbb{R}_ + ^2 $ . Under certain conditions on the functions g ₁, g ₂, we prove that the system of functions $g_{k,m} (x,\alpha^(n), h^(n)) (k,m \in \bar {\mathbb{Z}})$ , where $\alpha ^{\left( n \right)} ,h^{\left( n \right)} \in \mathbb{R}_ + ^2 $ are arbitrary infinitesimal sequences, is complete in the space C $\mathbb{R}^2 $ of uniformly continuous bounded functions f equipped with the norm $||f|| = \mathop {\sup }\limits_{x \in \mathbb{R}^2 } |f(x)|$ . Starting with the functions g _k,m , it is possible to construct a method for uniform approximating in $\mathbb{R}^2 $ any continuous function of bounded variation in the sense of Hardy. An error estimate is derived in terms of the second order moduli of continuity. Based on the obtained results, we discuss in detail the accuracy of uniform approximation of functions of several variables by linear functions. The error estimates are derived by using second order moduli of continuity. We pay a particular attention to sharpness of constants. Bibliography: 8 titles.