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.     

Orthogonality properties of linear combinations of orthogonal polynomials II

Let and be polynomials orthogonal on the unit circle with respect to the measures dσ and dμ, respectively. In this paper we consider the question how the orthogonality measures dσ and dμ are related to each other if the orthogonal polynomials are connected by a relation of the form , for , where . It turns out that the two measures are related by if , where and are known trigonometric polynomials of fixed degree and where the 's are the zeros of on . If the 's and 's are uniformly bounded then (under some additional conditions) much more can be said. Indeed, in this case the measures dσ and dμ have to be of the form and , respectively, where are nonnegative trigonometric polynomials. Finally, the question is considered to which weight functions polynomials of the form where denotes the reciprocal polynomial of , can be orthogonal. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

Nil-Manifolds Cannot be Immersed as Hypersurfaces in Euclidean Spaces

We prove that the $2n + 1$ -dimensional Heisenberg group H _n and the 4-manifolds $Nil^4 $ and $Nil^3 \times \mathbb{R}$ endowed with an arbitrary left-invariant metric admit no C ³-regular immersions into Euclidean spaces $\mathbb{R}^{2n + 2} $ and $\mathbb{R}^5 $ , respectively.     

Codimension 1 Subvarieties of {\mathcal{M}_g } and Real Gonality of Real Curves

Let ${\mathcal{M}_g }$ be the moduli space of smooth complex projective curves of genus g. Here we prove that the subset of ${\mathcal{M}_g }$ formed by all curves for which some Brill-Noether locus has dimension larger than the expected one has codimension at least two in ${\mathcal{M}_g }$ . As an application we show that if ${X \in \mathcal{M}_g }$ is defined over $\mathbb{R}$ then there exists a low degree pencil ${u:X \to \mathbb{P}^1 }$ defined over $\mathbb{R}.$      

Descriptions of exceptional sets in the circles for functions from the Bergman space

Let D be a domain in $\mathbb{C}^2 $ . For w ∈ $\mathbb{C}$ , let D_w=\{z \in $\mathbb{C}$ \, \vert \, (z,w)\in D\}. If f is a holomorphic and square-integrable function in D, then the set E(D, f) of all w such that f(., w) is not square-integrable in D _w is of measure zero. We call this set the exceptional set for f. In this note we prove that for every 0 < r < 1, and every G _δ-subset E of the circle C(0,r)=\{z \in $\mathbb{C}$ \, \vert \, \vert z \vert = r \},there exists a holomorphic square-integrable function f in the unit ball B in $\mathbb{C}$ ² such that E(B, f) = E.     

On sets of uniqueness for harmonic functions in the unit circle

The resutls of this paper show that the structure of sets mentioned in the title is not trivial. For example, it is shown that there exist countalbe sets of uniqueness for logarithmic potential, i.e., closed countable subsets E of the unit circle $\mathbb{T}$ such that $$f \in C(\mathbb{T}),f|_E = 0,U^f |_E = 0 \Rightarrow f \equiv 0.$$ Here $U^f (z) = \tfrac{1}{\pi }\int\limits_0^{2\pi } {f(e^{i\theta } )\log \tfrac{1}{{\left| {z - e^{i\theta } } \right|}}d\theta } $ . On the other hand, it is shown that every countable porous closed subset of $\mathbb{T}$ is a nonuniqueness set. Bibliography: 9 titles.     

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.     

On 1-isometries of affine quadrics over finite fields

Letq be a regular quadratic form on a vector space (V, $\mathbb{F}$ ) and assume $4 \leqslant dim V \leqslant \infty \wedge |\mathbb{F}| \in \mathbb{N}$ . A 1-isometry of the central quadric $\mathcal{F}: = \{ x \in V|q(x) = 1\}$ is a permutation ? of $\mathcal{F}$ such that (*) $$q(x - y) = \nu \Leftrightarrow q(x^\varphi - y^\varphi ) = \nu \forall x,y \in \mathcal{F}$$ holds true for a fixed element ν of $\mathbb{F}$ . For arbitraryν ∈ $\mathbb{F}$ we prove that? is induced (in a certain sense) by a semi-linear bijection $(\sigma ,\varrho ):(V,\mathbb{F}) \to (V,\mathbb{F})$ such thatq oσ =? oq, provided $\mathcal{F}$ contains lines and the exceptional case $(\nu = 2 \Lambda |\mathbb{F}| = 3 \Lambda \dim V = 4 \Lambda |\mathcal{F}| = 24)$ is excluded. In the exceptional case and as well in case of dim V = 3 there are counterexamples. The casesν ≠ 2 and v=2 require different techniques.     

Between\widehat{gl}(\infty ) and\widehat{sl}_N affine algebras. I. Geometrical actions

We consider the central extended $\widehat{gl}(\infty )$ Lie algebra and a set of its subalgebras parametrized by |q|=1, which coincides with the embedding of the quantum tori Lie algebras (QTLA) in $\widehat{gl}(\infty )$ . Forq ^N=1 there exists an ideal, and a factor over this ideal is isomorphic to an $\widehat{sl}_{N(z)} $ affine algebra. For a generic valueq the corresponding subalgebras are dense in $\widehat{gl}(\infty )$ . Thus, they interpolate between $\widehat{gl}(\infty )$ and $\widehat{sl}_{N(z)} $ . All these subalgebras are fixed points of automorphism of $\widehat{gl}(\infty )$ . Using the automorphisms, we construct geometrical actions for the subalgebras, starting from the Kirillov-Kostant form and the corresponding geometrical action for $\widehat{gl}(\infty )$ .     

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.     

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 $L_1 (\mathbb{T}^N )$ , $N \geqslant 2$ .     

An Algebraic Characterization of \mathcal{R} -Spaces

We study an algebraic structure naturally associated to a standard imbedding of an $\mathcal{R} $ -space. This structure determines completely the geometry of an $\mathcal{R} $ -space and reduces to a Jordan Triple System if the $\mathcal{R} $ -space is symmetric.     

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.     

Regular Semigroups of Endomorphisms of von Neumann Factors

We study a class of $E_0$ -semigroups of endomorphisms of a von Neumann factor $\mathcal{M}$ possessing the following property: an $e_0$ -semigroup of endomorphisms of $\mathcal{B}\left( \mathcal{H} \right)$ , where $\mathcal{H}$ is the standard representation space for $\mathcal{M}$ , and a product system of Hilbert spaces can be associated with each of these $E_0$ -semigroups.     

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.     

Estimates of the Number of Eigenvalues of Self-Adjoint Operator Functions

We consider an operator function F defined on the interval $\user2{[}\sigma \user2{,}\tau \user2{]} \subset \mathbb{R}$ whose values are semibounded self-adjoint operators in the Hilbert space $\mathfrak{H}$ . To the operator function F we assign quantities $\mathcal{N}_\user1{F}$ and ν _F (λ) that are, respectively, the number of eigenvalues of the operator function F on the half-interval [σ,τ) and the number of negative eigenvalues of the operator F(λ) for an arbitrary λ ∈ [σ,τ]. We present conditions under which the estimate $\mathcal{N}_\user1{F} \geqslant \nu _\user1{F} \user2{(}\tau \user2{)} - \nu _\user1{F} \user2{(}\sigma \user2{)}$ holds. We also establish conditions for the relation $\mathcal{N}_\user1{F} \geqslant \nu _\user1{F} \user2{(}\tau \user2{)} - \nu _\user1{F} \user2{(}\sigma \user2{)}$ to hold. The results obtained are applied to ordinary differential operator functions on a finite interval.     

Converse Quadrature sum Inequalities for Freud Weights. ii

Let $W: = \exp \left( { - Q} \right)$ , where $Q$ is of smooth polynomial growth at $\infty$ , for example $Q\left( x \right) = \left| x \right|^\beta ,\beta >1$ . We call $W^2 $ a Freud weight. Let $\left\{ {x_{j{\kern 1pt} n} } \right\}_{j = 1}^n $ and $\left\{ {\lambda _{j{\kern 1pt} n} } \right\}_{j = 1}^n $ denote respectively the zeros of the $n$ th orthonormal polynomial $p_n$ for $W^2 $ and the Christoffel numbers of order $n$ . We establish converse quadrature sum inequalities associated with W, such as $$\left\| {\left( {PW} \right)\left( x \right)\left( {1 + \left| x \right|} \right)^r } \right\|_{L_p \left( R \right)} $$ with $C$ independent of $n$ and polynomials P of degree $ < n$ , and suitable restrictions on $r$ , $R$ . We concentrate on the case ${ \geqq 4}$ , as the case ${p < 4}$ was handled earlier. We are able to treat a general class of Freud weights, whereas our earlier treatment dealt essentially with $\left( { - \left| x \right|^\beta } \right),\beta = 2,4,6,....$ Some applications to Lagrange interpolation are presented.     

Approximation of Dirichlet polynomials in cases of sparse exponents

Let \(0< \lambda \kappa \uparrow \infty ,\sum\nolimits_{\kappa = 1}^\infty {\lambda _\kappa ^{ - 1}< \infty } \) , and let γ be an analytic arc. For the Dirichlet polynomial \(P(z) = \sum\nolimits_1^n {a_k e^{\lambda _k .z} } \) , in angle \(E - \pi /2 + \varphi _0< \arg [ - (z - \alpha )]< \pi /2 + \varphi _0 ,0< \varphi _0< \pi /2,\operatorname{Re} \alpha< \beta = \mathop {\max }\limits_{t \in \gamma } \operatorname{Re} t\) we obtain the estimate $|P(z)|< A\mathop {\max }\limits_{t \in \gamma } |P(t)|$ where A depends only on angle E and {λ_k}. When γ is a segment, an estimate was obtained by L. Schwartz.