Uniform stability of the inverse Sturm-Liouville problem with respect to the spectral function in the scale of Sobolev spaces

We consider the inverse problem of recovering the potential for the Sturm-Liouville operator Ly = ?y″ + q(x)y on the interval [0, π] from the spectrum of the Dirichlet problem and norming constants (from the spectral function). For a fixed θ ≥ 0, with this problem we associate a map F: W ₂ ^θ → l _D ^θ , F(σ) = {s _k } ₁ ^∞ , where W ₂ ^θ = W ₂ ^θ [0, π] is the Sobolev space, σ = ∫ q is a primitive of the potential q ∈ W ₂ ^{θ ? 1} , and l _D ^θ is a specially constructed finite-dimensional extension of the weighted space l ₂ ^θ ; this extension contains the regularized spectral data s = {s _k } ₁ ^∞ for the problem of recovering the potential from the spectral function. The main result consists in proving both lower and upper uniform estimates for the norm of the difference ‖σ ? σ ₁‖ _θ in terms of the l _D ^θ norm of the difference of the regularized spectral data ‖s ? s₁‖ _θ . The result is new even for the classical case q ∈ L ₂, which corresponds to the case θ = 1.     

Interpolation in certain Hilbert spaces of analytic functions

In this article we study, for a Hilbert spaceB of analytic functions in the open unit disk, the dependence of the structure of the space of sequencesB(Z)={{f(z_k)} _k=1 ^∞ :f∈B} on the choice of the sequence Z={z_k} _k=1 ^∞ of distinct points of the unit disk [6].     

On the subfields of cyclotomic function fields

Let K = F(T) be the rational function field over a finite field of q elements. For any polynomial f(T) ∈ F [T] with positive degree, denote by Λ _f the torsion points of the Carlitz module for the polynomial ring F[T]. In this short paper, we will determine an explicit formula for the analytic class number for the unique subfield M of the cyclotomic function field K(Λ _P ) of degree k over F(T), where P ∈ F[T] is an irreducible polynomial of positive degree and k > 1 is a positive divisor of q ? 1. A formula for the analytic class number for the maximal real subfield M ⁺ of M is also presented. Futhermore, a relative class number formula for ideal class group of M will be given in terms of Artin L-function in this paper.     

A generalization of the katona theorem for crosst-intersecting families

LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:

1. There are two crosst-intersecting Hamming spheresA ₀,? ₀ with centerX such that |A| ≤ |A ₀| and|?| ≤ |? ₀| hold.
2. Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.

Using these claims, the following conjecture of Frankl is proven:

1. Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK _l ⁿ is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
2. Ifn + t = 2k then |A| |? ≤ (K _k ⁿ )² holds.

The extremal configurations are also determined.     

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).     

Jackson-type inequalities and widths of function classes in L 2

The sharp Jackson-type inequalities obtained by Taikov in the space L ₂ and containing the best approximation and the modulus of continuity of first order are generalized to moduli of continuity of kth order (k = 2, 3, ... ). We also obtain exact values of the n-widths of the function classes F(k, r, Φ) and F _k ^r (h), which are a generalization of the classes F(1, r, Φ) and F _k ^r (h) studied by Taikov.     

О раэложении мероморфных функций специального вида на простейщие дроби

The paper is devoted to study the entire functions L(λ) with simple real zeros λ_k, k = 1, 2, ..., that admit an expansion of Krein’s type: $$\frac{1}{{\mathcal{L}(\lambda )}} = \sum\limits_{k = 1}^\infty {\frac{{c_k }}{{\lambda - \lambda _k }}} ,\sum\limits_{k = 1}^\infty {\left| {c_k } \right| < \infty } .$$ We present a criterion for these expansions in terms of the sequence {L′ (λ _k )} _k=1 ^∞ . We show that this criterion is applicable to certain classes of meromorphic functions and make more precise a theorem of Sedletski? on the annihilating property in L ² systems of exponents.     

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) .     

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.     

On mappings preserving orthogonality of non-singular vectors

Letq be a regular quadratic form on a vector space (V,F) and letf be the bilinear form associated withq. Then, \(\dot V: = \{ z \in V|q(z) \ne 0\} \) is the set of non-singular vectors ofV, and forx, y ∈ \(\dot V\) , ?(x, y) ?f(x, y) ²/(q(x) · q(y)) is theq-measure of (x, y), where ?(x,y)=0 means thatx, y are orthogonal. For an arbitrary mapping \(\sigma :\dot V \to \dot V\) we consider the functional equations $$\begin{gathered} (I)\sphericalangle (x,y) = 0 \Leftrightarrow \sphericalangle (x^\sigma ,y^\sigma ) = 0\forall x,y \in \dot V, \hfill \\ (II)\sphericalangle (x,y) = \sphericalangle (x^\sigma ,y^\sigma )\forall x,y \in \dot V, \hfill \\ (III)f(x,y)^2 = f(x^\sigma ,y^\sigma )^2 \forall x,y \in \dot V, \hfill \\ \end{gathered} $$ and we state conditions on (V,F,q) such thatσ is induced by a mapping of a well-known type. In case of dimV ∈N?{0, 1, 2} ∧ ∣F∣ > 3, each of the assumptions (I), (II), (III) implies that there exist aρ-linear injectionξ :V →V and a fixed λ ∈F?{0} such thatF x ^σ =F x ^ξ ?x ∈ \(\dot V\) andf(x ^ξ,y ^ξ)=λ · (f(x, y))^ρ ?x, y ∈V. Moreover, (II) implies ρ =id _F ∧q(x ^ξ) = λ ·q(x) ?x ∈ \(\dot V\) , and (III) implies ρ=id _F ∧ λ ∈ {1,?1} ∧x ^σ ∈ {x ^ξ, ?x ^ξ} ?x ∈ \(\dot V\) . Other results obtained in this paper include the cases dimV = 2 resp. dimV ?N resp. ∣F∣ = 3.     

An embedding theorem for a weighted space of Sobolev type and correct solvability of the Sturm-Liouville equation

We consider the weighted space W ₁ ⁽²⁾ (?,q) of Sobolev type $$W_1^{(2)} (\mathbb{R},q) = \left\{ {y \in A_{loc}^{(1)} (\mathbb{R}):\left\| {y''} \right\|_{L_1 (\mathbb{R})} + \left\| {qy} \right\|_{L_1 (\mathbb{R})} < \infty } \right\} $$ and the equation $$ - y''(x) + q(x)y(x) = f(x),x \in \mathbb{R} $$ Here f ε L ₁(?) and 0 ? q ∈ L ₁ ^loc (?). We prove the following:

1. The problems of embedding W ₁ ⁽²⁾ (?q) ? L ₁(?) and of correct solvability of (1) in L ₁(?) are equivalent
2. an embedding W ₁ ⁽²⁾ (?,q) ? L ₁(?) exists if and only if $$\exists a > 0:\mathop {\inf }\limits_{x \in R} \int_{x - a}^{x + a} {q(t)dt > 0} $$

     

An analog of the weyl decomposition of the spaceL q (ω, ℝ n ) for a first-order differential operatorfor a first-order differential operator

We prove the following theorem: Suppose the function f(x) belongs toL _q (ω, ? ⁿ ), ω ? ? ^m , q∈(1, ∞), and satisfies the inequality $$|\int\limits_\omega {(f(x),{\mathbf{ }}v(x)){\mathbf{ }}dx| \leqslant \mu ||} v||'_q ,{\mathbf{ }}\tfrac{1}{q} + \tfrac{1}{{q'}} = 1,$$ for all n-dimensional vector-valued functions in the kernel of a scalar-valued first-order differential operator £ for which the second-order operatorLL ^* is elliptic. Then there exists a function p(x)∈W _q ¹ (ω) such that $$||f(x) - \mathfrak{L}^* p(x)||q \leqslant C_q \mu .$$ Bibliography: 6 titles.     

Minimal decompositions of graphs into mutually isomorphic subgraphs

SupposeG _n={G ₁, ...,G _k } is a collection of graphs, all havingn vertices ande edges. By aU-decomposition ofG _n we mean a set of partitions of the edge setsE(G _t ) of theG _i , sayE(G _t )== \(\sum\limits_{j = 1}^r {E_{ij} } \) E _ij , such that for eachj, all theE _ij , 1≦i≦k, are isomorphic as graphs. Define the functionU(G _n) to be the least possible value ofr aU-decomposition ofG _n can have. Finally, letU _k (n) denote the largest possible valueU(G) can assume whereG ranges over all sets ofk graphs havingn vertices and the same (unspecified) number of edges. In an earlier paper, the authors showed that $$U_2 (n) = \frac{2}{3}n + o(n).$$ In this paper, the value ofU _k (n) is investigated fork>2. It turns out rather unexpectedly that the leading term ofU _k (n) does not depend onk. In particular we show $$U_k (n) = \frac{3}{4}n + o_k (n),k \geqq 3.$$      

Wavelet estimations for density derivatives

Donoho et al. in 1996 have made almost perfect achievements in wavelet estimation for a density function f in Besov spaces Bsr,q(R). Motivated by their work, we define new linear and nonlinear wavelet estimators flin,nm, fnonn,m for density derivatives f(m). It turns out that the linear estimation E(‖flinn,m-f(m)‖p) for f(m) ∈ Bsr,q(R) attains the optimal when r≥ p, and the nonlinear one E(‖fnonn,m-f(m)‖p) does the same if r≤p/2(s+m)+1 . In addition, our method is applied to Sobolev spaces with non-negative integer exponents as well.     

Straight line arrangements in the real projective plane

Let A be an arrangement of n pseudolines in the real projective plane and let p ₃(A) be the number of triangles of A. Grünbaum has proposed the following question. Are there infinitely many simple arrangements of straight lines with p ₃(A)=1/3n(n?1)? In this paper we answer this question affirmatively.     

Criteria for the injectivity of analytic sheaves

It is shown that a moduleL over the sheafO of germs of holomorphic functions on a domain G of Cⁿ is injective if and only if the following conditions are satisfied; a)L is flabby; b) for every closed set S ?G and every point z λ G, the stalk se _z of the sheafS ^L;U1→Γ S ^(U:L) is an injectiveO _z -module. It follows in particular that the sheaf of germs of hyperfunctions is injective over the sheaf of germs of analytic functions.     

Singular integrals along Lipschitz curves with holomorphic kernels

If γ(x)=x+iA(x),tan ^?1‖A′‖_∞<ω<π/2,S _ω ⁰ ={z∈C}| |argz|<ω, or, |arg(-z)|<ω} We have proved that if φ is a holomorphic function in S _ω ⁰ and \(\left| {\varphi (z)} \right| \leqslant \frac{C}{{\left| z \right|}}\) , denotingT f (z)= ∫?(z-ζ)f(ζ)dζ, ?f∈C ₀(γ), ?z∈suppf, where C_c(γ) denotes the class of continuous functions with compact supports, then the following two conditions are equivalent:

1. T can be extended to be a bounded operator on L²(γ);
2. there exists a function ?₁ ∈H ^∞(S _ω ⁰ ) such that ?′₁(z)=?(z)+?(-z), ?z∈S _ω ⁰ ?z∈S _w ⁰ .

     

Asymptotic behaviour of the minima to a class of optimization problems for differential inclusions

Denoting byS _k (ξ _k ) the set of solutions of the Cauchy problem $\dot x \in F_k (t,x),x(0) = \xi _k $ , fork∈N∪{∞}, we prove that, under appropriate assumptions, the sequence {S _k (ξ _k )} _{k ∈ N} converges toS _∞(∈_∞) in the Kuratowski sense as well as in the Mosco sense. This result together with some facts from Γ-convergence theory are used to prove a result concerning the existence and the asymptotic behavior of the minima to the optimization problem $$\min \int_0^T {[g_k (t,x(t)) + h_k (t,\dot x(t))]} dt + \psi _k (\xi ),x \in S_k (\xi ),\xi \in K$$ withK a compact subset ofR ⁿ .     

k-Quasihyponormal operators are subscalar

In this paper we shall prove that if an operatorT∈L(⊕ ₁ ² H) is an operator matrix of the form $$T = \left( {\begin{array}{*{20}c} {T_1 } & {T_2 } \\ 0 & {T_3 } \\ \end{array} } \right)$$ whereT ₁ is hyponormal andT ₃ ^k =0, thenT is subscalar of order 2(k+1). Hence non-trivial invariant subspaces are known to exist if the spectrum ofT has interior in the plane as a result of a theorem of Eschmeier and Prunaru (see [EP]). As a corollary we get that anyk-quasihyponormal operators are subscalar.     

Best trigonometric and bilinear approximations of classes of functions of several variables

Order-sharp estimates of the best orthogonal trigonometric approximations of the Nikol’skii-Besov classes B _p,θ ^r of periodic functions of several variables in the space L _q are obtained. Also the orders of the best approximations of functions of 2d variables of the form g(x, y) = f(x?y), x, y ∈ $\mathbb{T}$ ^d = Π _j=1 ^d [?π, π], f(x) ∈ B _p,θ ^r , by linear combinations of products of functions of d variables are established.