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.     

On the number of non-isomorphic subgraphs

Let $\mathcal{K}$ be the family of graphs on ω₁ without cliques or independent subsets of sizew ₁. We prove that

1. it is consistent with CH that everyGε $\mathcal{K}$ has 2^ω many pairwise non-isomorphic subgraphs,
2. the following proposition holds in L: (*)there is a Gε $\mathcal{K}$ such that for each partition (A, B) of ω₁ either G?G[A] orG?G[B],
3. the failure of (*) is consistent with ZFC.

     

A Triangulation of ?P 3 as Symmetric Cube of S 2

The symmetric group $\operatorname{Sym}(d)$ acts on the Cartesian product (S ²) ^d by coordinate permutation, and the quotient space $(S^{2})^{d}/\operatorname{Sym}(d)$ is homeomorphic to the complex projective space ?P ^d . We used the case d=2 of this fact to construct a 10-vertex triangulation of ?P ² earlier. In this paper, we have constructed a 124-vertex simplicial subdivision $(S^{2})^{3}_{124}$ of the 64-vertex standard cellulation $(S^{2}_{4})^{3}$ of (S ²)³, such that the $\operatorname{Sym}(3)$ -action on this cellulation naturally extends to an action on $(S^{2})^{3}_{124}$ . Further, the $\operatorname{Sym}(3)$ -action on $(S^{2})^{3}_{124}$ is ??good??, so that the quotient simplicial complex $(S^{2})^{3}_{124}/\operatorname{Sym}(3)$ is a 30-vertex triangulation $\mathbb{C}P^{3}_{30}$ of ?P ³. In other words, we have constructed a simplicial realization $(S^{2})^{3}_{124} \to\mathbb{C} P^{3}_{30}$ of the branched covering (S ²)³???P ³.     

On permutations of central quadrics which preserve an inner distance

Letq be a regular quadratic form on a vector space (V, $\mathbb{F}$ ) and assume dimV ≥ 4 and ¦ $\mathbb{F}$ ¦ ≥ 4. We consider a permutation ? of the central affine quadric $\mathcal{F}$ := {x εV ¦q(x) = 1} such that $$(*)x \cdot y = \mu \Leftrightarrow x^\varphi \cdot y^\varphi = \mu \forall x,y\varepsilon \mathcal{F}$$ holds true, where μ is a fixed element of $\mathbb{F}$ and where “·” is the scalar product associated withq. We prove that ? is induced (in a certain sense) by a semi-linear bijection (σ,?): (V, $\mathbb{F}$ ) → (V, $\mathbb{F}$ ) such thatq o ?o q, provided $\mathcal{F}$ contains lines and the pair (μ, $\mathbb{F}$ ) has additional properties if there ar no planes in $\mathcal{F}$ . The cases μ,≠ 0 and μ = 0 require different techniques.     

Complete unitary invariant for some subnormal operator

IfS is a subnormal operator with minimal normal extensionN satisfying the conditions that (i) \(\left[ {S^* ,S} \right]^{\frac{1}{2}} \in \mathcal{L}^1\) , (ii) sp (S) is the unit disk and (iii) sp (N)={N: |z|=1 orz=a ₁,...,a _k then $$tr\left( {\left[ {S^* ,(\lambda I - S)^{ - 1} } \right]\left[ {S^* ,(\mu I - S)^{ - 1} } \right]} \right) = \frac{n}{{\lambda ^2 \mu ^2 }} + \sum\limits_{i,j = 1}^k {\frac{{\gamma ij}}{{\lambda \mu (\lambda - a_i )(\mu - a_j )}}} $$ . wheren=index ( \(S^* - \bar zI\) ) forz∈sp (S)/sp (N) and (γij) is a real symmetric matrix. The set {n, γij,i,j = 1,...,k} is a complete unitary invariant for an operator in the class of all irreducible subnormal operators satisfying conditions (i), (ii), (iii) and that there is at least one positive simple eigenvalue of [S ^*,S].     

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.     

О рядах по системе Фра нклина

In this paper some basis properties are proved for the series with respect to the Franklin system, which are analogous to those of the series with respect to the Haar system. In particular, the following statements hold:

1. The Franklin series \(\mathop \Sigma \limits_{n = 0}^\infty a_n f_n (x)\) converges a.e. onE if and only if \(\mathop \Sigma \limits_{n = 0}^\infty a_n^2 f_n^2 (x)< + \infty \) a.e. onE;
2. If the series \(\mathop \Sigma \limits_{n = 0}^\infty a_n f_n (x)\) , with coefficients ¦a _n ¦↓0, converges on a set of positive measure, then it is the Fourier-Franklin series of some function from \(\bigcap\limits_{p< \infty } {L_p } \) ;
3. The absolute convergence at a point for Fourier—Franklin series is a local property;
4. If an integrable function (fx) has a discontinuity of the first kind atx=x ₀, then its Fourier-Franklin series diverges atx=x ₀.

     

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.     

An inverse problem of the three-dimensional wave equation for a general annular vibrating membrane with piecewise smooth boundary conditions

This paper deals with the very interesting problem about the influence of piecewise smooth boundary conditions on the distribution of the eigenvalues of the negative Laplacian inR ³. The asymptotic expansion of the trace of the wave operator $\widehat\mu (t) = \sum\limits_{\upsilon = 1}^\infty {\exp \left( { - it\mu _\upsilon ^{1/2} } \right)} $ for small ?t? and $i = \sqrt { - 1} $ , where $\{ \mu _\nu \} _{\nu = 1}^\infty $ are the eigenvalues of the negative Laplacian $ - \nabla ^2 = - \sum\limits_{k = 1}^3 {\left( {\frac{\partial }{{\partial x^k }}} \right)} ^2 $ in the (x ¹,x ²,x ³), is studied for an annular vibrating membrane Ω inR ³ together with its smooth inner boundary surfaceS ₁ and its smooth outer boundary surfaceS ₂. In the present paper, a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth componentsS ^* _i(i=1, …,m) ofS ₁ and on the piecewise smooth componentsS ^* _i(i=m+1, …,n) ofS ₂ such that $S_1 = \bigcup\limits_{i = 1}^m {S_i^* } $ and $S_2 = \bigcup\limits_{i = m + 1}^n {S_i^* } $ are considered. The basic problem is to extract information on the geometry of the annular vibrating membrane ω from complete knowledge of its eigenvalues by analyzing the asymptotic expansions of the spectral function $\widehat\mu (t)$ for small ?t?.     

Subexponential loss rates in a GI/GI/1 queue with applications

Consider a single server queue with i.i.d. arrival and service processes, $\{ A,A_n ,n \geqslant 0\} $ and $\{ C,\;C_n ,n\;\; \geqslant \;\;0\} $ , respectively, and a finite buffer B. The queue content process $\{ Q_n^B ,n \geqslant 0\} $ is recursively defined as $Q_{n + 1}^B = \min ((Q_n^B + A_{n + 1} - C_{n + 1} )^ + ,B),\;\;q^ + = \max (0,q)$ . When $\mathbb{E}(A - C) < 0$ , and A has a subexponential distribution, we show that the stationary expected loss rate for this queue $E(Q_n^B + A_{n + 1} - C_{n + 1} - B)^ + $ has the following explicit asymptotic characterization: $${\mathbb{E}}\left( {Q_n^B + A_{n + 1} - C_{n + 1} - B} \right)^ + ~{\mathbb{E}}\left( {A - B} \right)^ + {as} B \to \infty ,$$ independently of the server process C _n . For a fluid queue with capacity c, M/G/∞ arrival process A _t , characterized by intermediately regularly varying on periods σ^on, which arrive with Poisson rate Λ, the average loss rate $\lambda _{{loss}}^B $ satisfies λ _loss ^B ～ Λ E(τ^onη — B)⁺ as B → ∞, where $\eta = r + \rho - c,\;\rho \; = \mathbb{E}A_t < \;\;c;r\;\;(c \leqslant r)$ is the rate at which the fluid is arriving during an on period. Accuracy of the above asymptotic relations is verified with extensive numerical and simulation experiments. These explicit formulas have potential application in designing communication networks that will carry traffic with long-tailed characteristics, e.g., Internet data services.     

Endpoint bounds for an analytic family of maximal functions

In this paper,for the plane curve T=.we define an analytic family of maximal functions asso-ciated to T asM_2f(λ)=sup_n>oh~-1∫_R相似文献   

On the limit superior of sequences of sets

qVЕРхНИИ пРЕДЕл пОслЕД ОВАтЕльНОстИ МНОжЕс тВA _n ОпРЕДЕльЕтсь сООтНО шЕНИЕМ \(\mathop {\lim sup}\limits_{n \to \infty } A_n = \mathop \cap \limits_{k = 1}^\infty \mathop \cup \limits_{n = k}^\infty A_n . B\) стАтьЕ РАссМАтРИВА Етсь слЕДУУЩИИ ВОпРО с: ЧтО МОжНО скАжАть О ВЕРхНИх пРЕДЕлАх \(\mathop {\lim sup}\limits_{k \to \infty } A_{n_k }\) , еслИ ИжВЕстНО, ЧтО пРЕсЕЧЕНИь \(\mathop \cap \limits_{k = 1}^\infty A_{n_k }\) «МАлы» Дль кАж-ДОИ пОДпОслЕДОВАтЕльНОстИ \((A_{n_k } )\) ? ДОкАжыВАЕтсь, Ч тО

1. ЕслИ \(\mathop \cap \limits_{k = 1}^\infty A_{n_k }\) — кОНЕЧНОЕ МНО жЕстВО Дль кАжДОИ пОДпОслЕДОВАтЕльНОстИ \((A_{n_k } )\) , тО НАИДЕтсь тАкАь пОДпО слЕДОВАтЕльНОсть, Дл ь кОтОРОИ МНОжЕстВО \(\mathop {\lim sup}\limits_{k \to \infty } A_{n_k }\) сЧЕтНО;
2. ЕслИ \(2^{\aleph _0 } = \aleph _1\) , тО сУЩЕстВУЕ т тАкАь пОслЕДОВАтЕл ьНОсть (A_n), ЧтО \(\mathop \cap \limits_{k = 1}^\infty A_{n_k }\) — сЧЕтНОЕ МНОжЕстВО Дль лУБОИ п ОДпОслЕДОВАтЕльНОстИ \((A_{n_k } )\) , НО \(\mathop {\lim sup}\limits_{k \to \infty } A_{n_k }\) ИМЕЕт МОЩ-НОсть кОНтИНУУМА;
3. ЕслИA _n — БОРЕлЕ ВскИЕ МНОжЕстВА В НЕкОтОРО М пОлНОМ сЕпАРАБЕльНО М МЕтРИЧЕскОМ пРОстРАНстВЕ, И \(\mathop \cap \limits_{k = 1}^\infty A_{n_k }\) — сЧЕт НОЕ МНОжЕстВО Дль кАж ДОИ пОДпОслЕДОВАтЕльНОстИ \((A_{n_k } )\) , тО сУЩЕстВУЕт тАкАь п ОДпОслЕДОВАтЕльНОсть, ЧтО \(\mathop {\lim sup}\limits_{k \to \infty } A_{n_k }\) — сЧЕтНОЕ МНОжЕстВО. кРОМЕ тОгО, ДОкАжАНО, Ч тО В слУЧАьх А) И В) В пОслЕДОВАтЕльНОстИ (A _n ) сУЩЕстВУЕт схОДьЩА ьсь пОДпОслЕДОВАтЕльНО сть.

кРОМЕ тОгО, ДОкАжАНО, Ч тО В слУЧАьх А) И В) В пОслЕДОВАтЕльНОстИ (А _n ) сУЩЕстВУЕт схОДьЩ Аьсь пОДпОслЕДОВАтЕльНО сть.     

On the integrability of radial basis functions

In this paper we investigate the integrability of certain radial basis functions. From the following forms of function σ, $$\varphi \left( r \right) = \left\{ \begin{gathered} \sum\limits_{k = 0}^{d + [a]} {c_k r^{a - k} + g(r) } r > A, \hfill \\ \sum\limits_{k = 0}^{d + [a]} {c_k r^{a - k} \ln r + g(r), } r > A. \hfill \\ \end{gathered} \right.$$ where A≧0 and $g \circ || \circ || \in L^1 \left( {R^d } \right)$ , we construct the function $$\psi (t) = \sum\limits_{j \in J} {a_j \varphi \left( {||t - t_j ||} \right),} $$ where J is a finite index set, $\left\{ {a_j } \right\}_{j \in J} \subseteq R$ and $\left\{ {t_j } \right\}_{j \in J} \subseteq R^d $ . We show that if $\hat \psi $ is continuous at the origin, the ψ is integrable in R^d.     

On Total Incomparability of Mixed Tsirelson Spaces

We give criteria of total incomparability for certain classes of mixed Tsirelson spaces. We show that spaces of the form $T\left[ {\left( {\mathcal{M}_k ,\theta _k } \right)_{k = 1}^l } \right]$ with index $i\left( {\mathcal{M}_k } \right)$ finite are either c ₀ or $\ell _p $ saturated for some p and we characterize when any two spaces of such a form are totally incomparable in terms of the index $i\left( {\mathcal{M}_k } \right)$ and the parameter θ _k . Also, we give sufficient conditions of total incomparability for a particular class of spaces of the form $T\left[ {\left( {\mathcal{A}_k ,\theta _k } \right)_{k = 1}^\infty } \right]$ in terms of the asymptotic behaviour of the sequence $\left\| {\sum\limits_{j = 1}^n {e_i } } \right\|$ where (e _i is the canonical basis.     

Subspace Arrangements of Type B n and D n

Let ${\mathcal{D}}_{n,k} $ be the family of linear subspaces of ?ⁿ given by all equations of the form $\varepsilon _1 x_{i_1 } = \varepsilon _2 x_{i_2 } = \cdot \cdot \cdot \varepsilon _k x_{i_k } ,$ for 1 ≤ < ? ? ? < i _k ≤ i and $\left( {\varepsilon _1 ,...,\varepsilon _k } \right)\varepsilon \left\{ { + 1, - 1} \right\}^k $ Also let ${\mathcal{B}}_{n,k,h} $ be ${\mathcal{D}}_{n,k} $ enlarged by the subspaces $x_{j_1 } = x_{j_2 } = \cdot \cdot \cdot x_{j_h } = 0,$ for 1 ≤. The special cases ${\mathcal{B}}_{n,2,1} $ and ${\mathcal{D}}_{n,2} $ are well known as the reflection hyperplane arrangements corresponding to the Coxeter groups of type B _nand D _n respectively. In this paper we study combinatorial and topological properties of the intersection lattices of these subspace arrangements. Expressions for their Möbius functions and characteristic polynomials are derived. Lexicographic shellability is established in the case of ${\mathcal{B}}_{n,k,h,} 1 \leqslant h < k$ , which allows computation of the homology of its intersection lattice and the cohomology groups of the manifold $\begin{gathered} {\mathcal{D}}_{n,2} \\ M_{n,k,h,} = {\mathbb{R}}^n \backslash \bigcup {{\mathcal{B}}_{n,k,h,} } \\ \end{gathered} $ . For instance, it is shown that $H^d \left( {M_{n,k,k - 1} } \right)$ is torsion-free and is nonzero if and only if d = t(k ? 2) for some $t,0 \leqslant t \leqslant \left[ {{n \mathord{\left/ {\vphantom {n k}} \right. \kern-0em} k}} \right]$ . Torsion-free cohomology follows also for the complement in ?ⁿof the complexification ${\mathcal{B}}_{n,k,h}^C ,1 \leqslant h < k$ .     

The second variational formula for Willmore submanifolds in Sn

In [17] the third author presented Moebius geometry for sub-manifolds in Sⁿ and calculated the first variational formula of the Willmore functional by using Moebius invariants. In this paper we present the second variational formula for Willmore submanifolds. As an application of these variational formulas we give the standard examples of Willmore hypersurfaces $ \lbrace W_{k}^{m}:= S^{k}(\sqrt {(m-k)/m}) \times S^{m-k}(\sqrt {k/m}), 1 \leq k \leq m-1 \rbrace $ in S^m+1 (which can be obtained by exchanging radii in the Clifford tori $ S^{k}(\sqrt {k/m}) \times S^{m-k}(\sqrt {(m-k)/m)})$ and show that they are stable Willmore hypersurfaces. In case of surfaces in S³, the stability of the Clifford torus $ S^{1}{({1\over \sqrt {2}})}\times S^{1}{({1\over \sqrt {2}})} $ was proved by J. L. Weiner in [18]. We give also some examples of m-dimensional Willmore submanifolds in an n-dimensional unit sphere Sⁿ.     

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 .