Приближение всплесками и поперечники Фурье классов периодических функций многих переменных. II

Estimates sharp in order for Fourier widths of the classes $ B_{pq}^{sm} (\mathbb{T}^k ) $ and $ L_{pq}^{sm} (\mathbb{T}^k ) $ of Nikol??skii-Besov and Lizorkin-Triebel types, respectively, in the space $ L_r (\mathbb{T}^k ) $ are established for a certain range of the parameters s, p, q, r (here s ?? (0,??) ⁿ , 1 ??p, r, q ???, 1 ?? n ?? k, m = (m ₁, ??,m _n ) ?? ? ⁿ : m ₁ + ?? + m _n = k).     

Estimates for wave and Klein-Gordon equations on modulation spaces

We prove that the fundamental semi-group eit(m 2I+|Δ|)1/2(m = 0) of the Klein-Gordon equation is bounded on the modulation space M ps,q(Rn) for all 0 < p,q ∞ and s ∈ R.Similarly,we prove that the wave semi-group eit|Δ|1/2 is bounded on the Hardy type modulation spaces μsp,q(Rn) for all 0 < p,q ∞,and s ∈ R.All the bounds have an asymptotic factor tn|1/p 1/2| as t goes to the infinity.These results extend some known results for the case of p 1.Also,some applications for the Cauchy problems related to the semi-group eit(m2I+|Δ|)1/2 are obtained.Finally we discuss the optimum of the factor tn|1/p 1/2| and raise some unsolved problems.     

Nonlinear approximations of classes of periodic functions of many variables

Order-sharp estimates are established for the best N-term approximations of functions in the classes $B_{pq}^{sm} (\mathbb{T}^k )$ and $L_{pq}^{sm} (\mathbb{T}^k )$ of Nikol’skii-Besov and Lizorkin-Triebel types with respect to the multiple system of Meyer wavelets in the metric of $L_r (\mathbb{T}^k )$ for various relations between the parameters s, p, q, r, and m (s = (s ₁, ..., s _n ) ∈ ? ₊ ⁿ , 1 ≤ p, q, r ≤ ∞, m = (m ₁, ..., m _n ) ∈ ? ⁿ , and k = m ₁ + ... + m _n ). The proof of upper estimates is based on variants of the so-called greedy algorithms.     

Gaussian decompositions in function spaces

We introduce Gausslets , where P(x) are distinguished polynomials in ?ⁿ. Combined with dilations x?2 ^v x, where v∈N₀, and translations x?x+m, where m ∈ ?ⁿ, one obtains frames in the function spaces B _pq ^s (? ⁿ ) and F _pq ^s (? ⁿ ) for all possible parameters ^{s, p}, and ^q.     

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 Davenport constant and on the structure of extremal zero-sum free sequences

Let $G = C_{n_1 } \oplus \cdots \oplus C_{n_r }$ with 1 < n ₁ | ?? | n _r be a finite abelian group, d*(G) = n ₁ +??+n _r ?r, and let d(G) denote the maximal length of a zerosum free sequence over G. Then d(G) ?? d*(G), and the standing conjecture is that equality holds for G = C _n ^r . We show that equality does not hold for C ₂ ?? C _2n ^r , where n ?? 3 is odd and r ?? 4. This gives new information on the structure of extremal zero-sum free sequences over C _2n ^r .     

The proof of a conjecture concerning the intersection of k-generalized Fibonacci sequences

For k ≥ 2, the k-generalized Fibonacci sequence (F _n ^(k) ) _n is defined by the initial values 0, 0, …, 0,1 (k terms) and such that each term afterwards is the sum of the k preceding terms. In 2005, Noe and Post conjectured that the only solutions of Diophantine equation F _m ^(k) = F _n ^(?) , with ? > k > 1, n > ? + 1, m > k + 1 are $(m,n,\ell ,k) = (7,6,3,2)and(12,11,7,3)$ . In this paper, we confirm this conjecture.     

On the geometry of flat pseudo-Riemannian homogeneous spaces

Let M = ? _s ⁿ /Γ be a complete flat pseudo-Riemannian homogeneous manifold, Γ ? Iso(? _s ⁿ ) its fundamental group and G the Zariski closure of Γ in Iso(? _s ⁿ ). We show that the G-orbits in ? _s ⁿ are affine subspaces and affinely diffeomorphic to G endowed with the (0)-connection. If the restriction of the pseudo-scalar product on ? _s ⁿ to the G-orbits is nondegenerate, then M has abelian linear holonomy. If additionally G is not abelian, then G contains a certain subgroup of dimension 6. In particular, for non-abelian G, orbits with non-degenerate metric can appear only if dim G ≥ 6. Moreover, we show that ? _s ⁿ is a trivial algebraic principal bundle G → M → ? ^n?k . As a consquence, M is a trivial smooth bundle G/Γ → M → ? ^n?k with compact fiber G/Γ.     

Inversion complexity of self-correcting circuits for a certain sequence of boolean functions

The asymptotics L _k ^? (f ₂ ⁿ ) ?? n min{k+1, p} is obtained for the sequence of Boolean functions $f_2^n \left( {x_1 , \ldots ,x_n } \right) = \mathop \vee \limits_{1 \leqslant i < j \leqslant n}$ for any fixed k, p ?? 1 and growing n, here L _k ^? (f ₂ ⁿ ) is the inversion complexity of realization of the function f ₂ ⁿ by k-self-correcting circuits of functional elements in the basis B = {&, ?}, p is the weight of a reliable invertor.     

The inverse eigenvalue problem of reflexive matrices with a submatrix constraint and its approximation

In this paper, we first consider the existence of and the general expression for the solution to the constrained inverse eigenvalue problem defined as follows: given a generalized reflection matrix P∈R ^n×n , a set of complex n-vectors {x _i } _i=1 ^m , a set of complex numbers {λ _i } _i=1 ^m , and an s-by-s real matrix C ₀, find an n-by-n real reflexive matrix C such that the s-by-s leading principal submatrix of C is C ₀, and {x _i } _i=1 ^m and {λ _i } _i=1 ^m are the eigenvectors and eigenvalues of C, respectively. We are then concerned with the best approximation problem for the constrained inverse problem whose solution set is nonempty. That is, given an arbitrary real n-by-n matrix $\tilde{C}$ , find a matrix C which is the solution to the constrained inverse problem such that the distance between C and $\tilde{C}$ is minimized in the Frobenius norm. We give an explicit solution and a numerical algorithm to the best approximation problem. An illustrative experiment is also presented.     

Sebestyén's Moment Problem And Regular Dilations

We give a necessary and sufficient condition for a double indexed sequence {h _m ⁿ } of vectors in a Hilbert space such that it can be represented in the form h _m ⁿ = T ^m S ⁿ h ₀ ⁰ , (?)m, n ∈ N, where (T,S) is a pair of commuting contractions having regular unitary dilation.     

The reconstruction problem for certain infinite graphs

We are concerned with the notion of the degree-type (D _G ⁱ )_i∈ω of a graphG, whereD _G ⁱ is defined to be the number of vertices inG with degreei. In the first section the following results are proven:

1. IfG is a connected, locally finite, countably infinite graph such that there exists ani so thatD _G ⁱ andD _G ⁱ⁺¹ are both finite and different from 0, thenG is reconstructible.
2. Locally finite, countably infinite graphsG, for which infinitely manyD _G ⁱ are different from 0 but only finitely manyD _G ⁱ are infinite, are reconstructible.

In the second section we give some results about the reconstructibility of certain locally finite countably infinite interval graphs and show that a reconstruction of a planar, infinite graph has to be planar too.     

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.     

The limits of determinacy in second order arithmetic: consistency and complexity strength

We prove that determinacy for all Boolean combinations of \({F_{\sigma \delta }}\) (Π ₃ ⁰ ) sets implies the consistency of second-order arithmetic and more. Indeed, it is equivalent to the statement saying that for every set X and every number n, there exists a β-model of Π _n ¹ -comprehension containing X. We prove this result by providing a careful level-by-level analysis of determinacy at the finite level of the difference hierarchy on \({F_{\sigma \delta }}\) (Π ₃ ⁰ ) sets in terms of both reverse mathematics, complexity and consistency strength. We show that, for n ≥ 1, determinacy for sets at the nth level in this difference hierarchy lies strictly between (in the reverse mathematical sense of logical implication) the existence of β-models of Π _n+2 ¹ -comprehension containing any given set X, and the existence of β-models of Δ _n+2 ¹ -comprehension containing any given set X. Thus the nth of these determinacy axioms lies strictly between Π _n+2 ¹ -comprehension and Δ _n+2 ¹ -comprehension in terms of consistency strength. The major new technical result on which these proof theoretic ones are based is a complexity theoretic one. The nth determinacy axiom implies closure under the operation taking a set X to the least Σ _n+1 admissible containing X (for n = 1; this is due to Welch [9]).     

Laplace-Transformation nichtnegativer und vektorwertiger Maße

Calling a function f: R ₊ ^p →R with $$\sum\limits_{i = 1}^n { \sum\limits_{j = 1}^n {\alpha _i \alpha _j } f(t_i + t_j )} \geqslant 0 for all (\alpha _1 ,...,\alpha _n ) \varepsilon R^n ,$$ , (t₁,...,t_n∈(R ₊ ^p )ⁿ, n∈? positiv semidefinit, the Laplace-transformations of finite nonnegative measures on R ₊ ^p are charac terised as the continuous bounded positiv semidefinit functions. Let H be a real Hilbertspace. A σ-additive mapping \(M: \mathfrak{B}_ + ^p \to H\) is called an orthogonal measure iff 〈M(A), M(B)?=0 for A∩B=ø. Exactly those mappings Y: R ₊ ^p →H are Laplacetransformations of H-valued orthogonal measures, which are continuous and bounded and for which ?Y(s), Y(t)? is only a function of s + t. Using this result one obtains a representation theorem for continuous semi-grouphomomorphisms defined on R ₊ ^p with values in the “unit intervall” of the selfadjoint operators on H.     

Caricature of hydrodynamics for lattice dynamics

We consider the lattice dynamics in the half-space, with zero boundary condition. The initial data are supposed to be random function. We introduce the family of initial measures {?? ₀ ^? , ? > 0} depending on a small scaling parameter ?. We assume that the measures ?? ₀ ^? are locally homogeneous for space translations of order much less than ? ^?1 and nonhomogeneous for translations of order ? ^?1. Moreover, the covariance of ?? ₀ ^? decreases with distance uniformly in ?. Given ?? ?? ? / 0, r ?? ? ₊ ^d , and ?? > 0, we consider the distributions of random solution in the time moments t = ??/? _?? and at lattice points close to [r/?] ?? ? ₊ ^d . Themain goal is to study the asymptotic behavior of these distributions as ? ?? 0 and to derive the limit hydrodynamic equations of the Euler or Navier-Stokes type.     

Linearity and Second Fundamental Forms for Proper Holomorphic Maps from \mathbb{B}^{n+1} to \mathbb{B}^{4n-3}

For a holomorphic proper map F from the ball $\mathbb{B}^{n+1}$ into $\mathbb{B}^{N+1}$ that is C ³ smooth up to the boundary, the image $M=F(\partial\mathbb{B}^{n})$ is an immersed CR submanifold in the sphere $\partial \mathbb{B}^{N+1}$ on which some second fundamental forms II _M and $\mathit{II}^{CR}_{M}$ can be defined. It is shown that when 4??n+1<N+1??4n?3, F is linear fractional if and only if $\mathit{II}_{M} - \mathit{II}_{M}^{CR} \equiv 0$ .     

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 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 certain estimates for orthogonal polynomials depending on parameters

Рассматривается сис тема ортогональных м ногочленов {P _n (z)} ₀ ^∞ , удовлетворяющ их условиям $$\frac{1}{{2\pi }}\int\limits_0^{2\pi } {P_m (z)\overline {P_n (z)} d\sigma (\theta ) = \left\{ {\begin{array}{*{20}c} {0,m \ne n,P_n (z) = z^n + ...,z = \exp (i\theta ),} \\ {h_n > 0,m = n(n = 0,1,...),} \\ \end{array} } \right.} $$ где σ (θ) — ограниченная неу бывающая на отрезке [0,2π] функция с бесчисленным множе ством точек роста. Вводится последовательность параметров {а_{n 0} ^∞ , независимых дру г от друга и подчиненных единств енному ограничению { ¦а_n¦<1} ₀ ^∞ ; все многочлены {Р _n (z)} 0/∞ можно найти по формуле $$P_0 = 1,P_{k + 1(z)} = zP_k (z) - a_k P_k^ * (z),P_k^ * (z) = z^k \bar P_k \left( {\frac{1}{z}} \right)(k = 0,1,...)$$ . Многие свойства и оце нки для {P _n (z)} ₀ ^∞ и (θ) можн о найти в зависимости от этих параметров; например, условие \(\mathop \Sigma \limits_{n = 0}^\infty \left| {a_n } \right|^2< \infty \) , бо лее общее, чем условие Г. Cerë, необходимо и достато чно для справедливости а симптотической форм улы в области ¦z¦>1. Пользуясь этим ме тодом, можно найти также реш ение задачи В. А. Стекло ва.