Periodicity of the Spectrum of a Finite Union of Intervals

A set Ω, of Lebesgue measure 1, in the real line is called spectral if there is a set Λ of real numbers such that the exponential functions e _λ (x)=exp (2πiλx), λ∈Λ, form a complete orthonormal system on L ²(Ω). Such a set Λ is called a spectrum of Ω. In this note we present a simplified proof of the fact that any spectrum Λ of a set Ω which is finite union of intervals must be periodic. The original proof is due to Bose and Madan.     

Hardy spaces via distribution spaces

Let ℐ(ℝⁿ) be the Schwartz class on ℝⁿ and ℐ_∞(ℝⁿ) be the collection of functions ϕ ∊ ℐ(ℝⁿ) with additional property that

Central polynomials and matrix invariants

LetK be a field, charK=0 andM _n (K) the algebra ofn×n matrices overK. If λ=(λ₁,…,λ _m ) andμ=(μ ₁,…,μ _m ) are partitions ofn ² let wherex ₁,…,x _n ²,y ₁,…,y _n ² are noncommuting indeterminates andS _n ² is the symmetric group of degreen ². The polynomialsF ^{λ, μ} , when evaluated inM _n (K), take central values and we study the problem of classifying those partitions λ,μ for whichF ^{λ, μ} is a central polynomial (not a polynomial identity) forM _n (K). We give a formula that allows us to evaluateF ^{λ, μ} inM(K) in general and we prove that if λ andμ are not both derived in a suitable way from the partition δ=(1, 3,…, 2n−3, 2n−1), thenF ^{λ, μ} is a polynomial identity forM _n (K). As an application, we exhibit a new class of central polynomials forM _n (K). In memory of Shimshon Amitsur Research supported by a grant from MURST of Italy.     

Poisson Type Generators for L 1(ℝ)

We characterize the discrete sets Λ⊆ℝ such that {φ(t−λ),λ∈Λ} span L ¹(ℝ), φ being an L ¹(ℝ)-function whose Fourier transform behaves like e ^−2π|ξ|.     

Geometric minimality conditions for systems of exponentials inL p [−π, π]

A proof is given of the stability theorem for minimal systems of exponentialse(Λ) = {e ^iλx }λ∈Λ inL _p [−π, π], where Λ ⊂ ℂ is a discrete subset. Geometric minimality conditions for such systems are obtained. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 773–777, November, 1995. I wish to express gratitude to A. A. Shkalikov, who posed the problem and paid constant attention to this work.     

Existence of limits of maximal means

For the class II(ℝ ^m ) of continuous almost periodic functionsf: ℝ ^m → ℝ, we consider the problem of the existence of the limit

On approximation of periodic functions by Fourier sums

Let L _p , 1 ≤ p< ∞, be the space of 2π-periodic functions f with the norm || f ||_p = ( ò _{- p}^p | f |^p )^{1 \mathord}

Dual partially harmonic tensors and Brauer-Schur-Weyl duality

Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let $ \mathfrak{B}_n^{(f)} Let V be a 2m-dimensional symplectic vector space over an algebraically closed field K. Let \mathfrakB_n^(f) \mathfrak{B}_n^{(f)} be the two-sided ideal of the Brauer algebra \mathfrakB_n( - 2m ) {\mathfrak{B}_n}\left( { - 2m} \right) over K generated by e ₁ e _3⋯ e _2f-1 where 0 ≤ f ≤ [n/2]. Let HT_f ^?n \mathcal{H}\mathcal{T}_f^{ \otimes n} be the subspace of partial-harmonic tensors of valence f in V ^⊗n . In this paper we prove that dimHT_f ^?n \mathcal{H}\mathcal{T}_f^{ \otimes n} and dim \textEn\textd_K\textSp(V)( V ^?n \mathord

Toeplitz operators related to strongly singular Calderón-Zygmund operators

In this paper, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε (ℝⁿ) is discussed from L ^p(ℝⁿ) to L ^q(ℝⁿ), , and from L ^p(ℝⁿ) to Triebel-Lizorkin space . We also obtain the boundedness of generalized Toeplitz operator Θ _α0 ^b from L ^p(ℝⁿ) to L ^q(ℝⁿ), . All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator T _b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L ^p(ℝⁿ), 1 < p < ∞.     

Hausdorff dimension of Julia set of a polynomial with a Siegel disk

Let f(z) = e2πiθz(1 z/d)d,θ∈R\Q be a polynomial. Ifθis an irrational number of bounded type, it is easy to see that f(z) has a Siegel disk centered at 0. In this paper, we will show that the Hausdorff dimension of the Julia set of f(z) satisfies Dim(J(f))<2.     

A continuous extension of the de la Vallée Poussin means

Among the many interesting results of their 1958 paper, G. Pólya and I. J. Schoenberg studied the de la Vallée Poussin means of analytic functions. These are polynomial approximations of a given analytic function on the unit disk obtained by taking Hadamard products of the functionf with certain polynomialsV _n (z), wheren is the degree of the polynomial. The polynomial approximationsV _n *f converge locally uniformly tof asn→∞. In this paper, we define a subordination chainV _λ (z),γ>0, |z|<1, of convex mappings of the disk that for integer values is the same as the previously definedV _n (z). Iff is a conformal mapping of the diskD onto a convex domain, thenV _λ *f→f locally uniformly as λ→∞, and in fact when λ₂ > λ₁. We also consider Hadamard products of theV _λ with complex-valued harmonic mappings of the disk. This work was supported by the Volkswagen Stiftung (RiP-program at Oberwolfach). S. R. received partial support also from INTAS (Project 99-00089) and the German-Israeli Foundation (grant G-643-117.6/1999).     

Completeness of systems of entire functions in spaces of holomorphic functions

Letf be an entire function in . For a broad class of distribution densities of the set Λ, a scale of sufficient conditions for the completeness of the system of functions {f(λ×z):λ∈Λ},z∈E, where , in the spaceH(E) of holomorphic functions onE with respect to the topology of uniform convergence on compact subsets is given in terms of the mutual indicator of the functionf and the setE. These conditions are new already forn=1 even ifE is a disk. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 603–616, October, 1999.     

Euler operator and homogeneous hida distributions

Let (S)⊄L ²(S′(∔),μ)⊄(S)^* be the Gel'fand triple over the white noise space (S′(∔),μ). Let (e _n ,n>-0) be the ONB ofL ²(∔) consisting of the eigenfunctions of the s.a. operator . In this paper the Euler operator Δ _E is defined as the sum , where ∂ _i stands for the differential operatorD _{e i}. It is shown that Δ _E is the infinitesimal generator of the semigroup (T _t ), where (T _t ϕ)(x)=ϕ(e ^t x) for ϕ∈(S). Similarly to the finite dimensional case, the λ-order homogeneous test functionals are characterized by the Euler equation: Δ _Eϕ =λ_ϕ. Via this characterization the λ-order homogeneous Hida distributions are defined and their properties are worked out. Supported by the National Natural Science Foundation of China.     

On the stability of the uniform minimality of a set of exponentials

Some conditions on sequences (λ _n ) and (μ _n ) to be nearby are given in order that the corresponding systems of complex exponentials (exp(iλ _n t)) and (exp(iμ _n t)) be simultaneously uniformly minimal in L ^p (−π, π), 1 ≤ p < ∞, and in C[−π, π]. __________ Translated from Sovremennaya Matematika. Fundamental'nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 25, Theory of Functions, 2007.     

Comonotone approximation of twice differentiable periodic functions

In the case where a 2π-periodic function f is twice continuously differentiable on the real axis ℝ and changes its monotonicity at different fixed points y _i ∈ [− π, π), i = 1,…, 2s, s ∈ ℕ (i.e., on ℝ, there exists a set Y := {y _i } _i∈ℤ of points y _i = y _i+2s + 2π such that the function f does not decrease on [y _i , y _i−1] if i is odd and does not increase if i is even), for any natural k and n, n ≥ N(Y, k) = const, we construct a trigonometric polynomial T _n of order ≤n that changes its monotonicity at the same points y _i ∈ Y as f and is such that

A Characterization of Quadric Constant Mean Curvature Hypersurfaces of Spheres

Let be an immersion of a complete n-dimensional oriented manifold. For any v∈ℝ ⁿ⁺², let us denote by ℓ _v :M→ℝ the function given by ℓ _v (x)=〈φ(x),v〉 and by f _v :M→ℝ, the function given by f _v (x)=〈ν(x),v〉, where is a Gauss map. We will prove that if M has constant mean curvature, and, for some v≠0 and some real number λ, we have that ℓ _v =λ f _v , then, φ(M) is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface M ⁿ in which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to 2n+4. A. Brasil Jr. was partially supported by CNPq, Brazil, 306626/2007-1.     

Notes on combinatorial set theory

We shall prove some unconnected theorems: (1) (G.C.H.) \omega _{\alpha + 1} \to \left( {\omega _\alpha + \xi } \right)_2^2 when ℵ_α is regular, │ξ│⁺<ωα. (2) There is a Jonsson algebra in ℵ_α+n, and \aleph _{a + n} \not \to \left[ {\aleph _{a + n} } \right]_{\aleph _{a + n} }^{n + 1} if 2^{\aleph _{ - - } } = \aleph _{a + n} \cdot (3) If λ>ℵ₀ is a strong limit cardinal, then among the graphs with ≦λ vertices each of valence <λ there is a universal one. (4)(G.C.H.) If f is a set mapping on \omega _{a + 1} (ℵ_α regular) │f(x)∩f(y│<ℵ_α, then there is a free subset of order-type ζ for every ζ<ω_α+1.     

Dilation of Newton Polytope and <Emphasis Type="Italic">p</Emphasis>-adic Estimate

Let f(X) be a polynomial in n variables over the finite field  \mathbbF_q\mathbb{F}_{q}. Its Newton polytope Δ(f) is the convex closure in ℝ ⁿ of the origin and the exponent vectors (viewed as points in ℝ ⁿ ) of monomials in f(X). The minimal dilation of Δ(f) such that it contains at least one lattice point of $\mathbb{Z}_{>0}^{n}$\mathbb{Z}_{>0}^{n} plays a vital pole in the p-adic estimate of the number of zeros of f(X) in  \mathbbF_q\mathbb{F}_{q}. Using this fact, we obtain several tight and computational bounds for the dilation which unify and improve a number of previous results in this direction.     

Existence and analyticity of eigenvalues of a two-channel molecular resonance model

We consider a family of operators H_γμ(k), k ∈ \mathbbT^d \mathbb{T}^d := (−π,π]^d, associated with the Hamiltonian of a system consisting of at most two particles on a d-dimensional lattice ℤ^d, interacting via both a pair contact potential (μ > 0) and creation and annihilation operators (γ > 0). We prove the existence of a unique eigenvalue of H_γμ(k), k ∈ \mathbbT^d \mathbb{T}^d , or its absence depending on both the interaction parameters γ,μ ≥ 0 and the system quasimomentum k ∈ \mathbbT^d \mathbb{T}^d . We show that the corresponding eigenvector is analytic. We establish that the eigenvalue and eigenvector are analytic functions of the quasimomentum k ∈ \mathbbT^d \mathbb{T}^d in the existence domain G ⊂ \mathbbT^d \mathbb{T}^d .