Characterization of Scattering Data for the AKNS System

We characterize the scattering data of the AKNS system with vanishing boundary conditions. We prove a 1,1-correspondence between L ¹-potentials without spectral singularities and Marchenko integral kernels which are sums of an L ¹ function (having a reflection coefficient as its Fourier transform) and a finite exponential sum encoding bound states and norming constants. We give characterization results in the focusing and defocusing cases separately.     

OnL 1-convergence of modified complex trigonometric sums

We study hereL ¹-convergence of a complex trigonometric sum and obtain a new necessary and sufficient condition for theL ¹-convergence of Fourier series.     

On Jodeit’s multiplier extension theorems

Let Δ(x) = max {1 - ¦x¦, 0} for all x ∈ ?, and let ξ_[0,1) be the characteristic function of the interval 0 ≤x < 1. Two seminal theorems of M. Jodeit assert that A and ξ_[0,1) act as summability kernels convertingp-multipliers for Fourier series to multipliers forL ^P (?). The summability process corresponding to Δ extendsL ^P (T)-multipliers from ? to ? by linearity over the intervals [n, n + 1],n ∈ ?, when 1 ≤p < ∞, while the summability process corresponding to ξ_[0,1) extends L^P(T)-multipliers by constancy on the intervals [n, n + 1),n ∈ ?, when 1 <p < ∞. We describe how both these results have the following complete generalization: for 1 ≤p < ∞, an arbitrary compactly supported multiplier forL ^P (?) will act as a summability kernel forL ^P (T)-multipliers, transferring maximal estimates from L^P(T) to L^P(?). In particular, specialization of this maximal theorem to Jodeit’s summability kernel ξ_{[0, 1)} provides a quick structural way to recover the fact that the maximal partial sum operator on L^P(?), 1 <p < ∞, inherits strong type (p,p)-boundedness from the Carleson-Hunt Theorem for Fourier series. Another result of Jodeit treats summability kernels lacking compact support, and we show that this aspect of multiplier theory sets up a lively interplay with entire functions of exponential type and sampling methods for band limited distributions.     

The L p -L q analog of Morgan’s theorem on exponential solvable Lie groups

In this paper, we define an analog of the L ^p -L ^q Morgan’s uncertainty principle for any exponential solvable Lie group G (p, q ∈ [1,+∞]). When G is nilpotent and has a noncompact center, the proof of such an analog is given for p, q ∈ [2,+∞], extending the earlier settings ([2], [4] and [5]). Such a result is only known for some particular restrictive cases so far. We also prove the result for general exponential Lie groups with nontrivial center.     

Tight bounds for mixing of the Swendsen–Wang algorithm at the Potts transition point

We study two widely used algorithms for the Potts model on rectangular subsets of the hypercubic lattice ${\mathbb{Z}^{d}}$ —heat bath dynamics and the Swendsen–Wang algorithm—and prove that, under certain circumstances, the mixing in these algorithms is torpid or slow. In particular, we show that for heat bath dynamics throughout the region of phase coexistence, and for the Swendsen–Wang algorithm at the transition point, the mixing time in a box of side length L with periodic boundary conditions has upper and lower bounds which are exponential in L ^d-1. This work provides the first upper bound of this form for the Swendsen–Wang algorithm, and gives lower bounds for both algorithms which significantly improve the previous lower bounds that were exponential in L/(log L)².     

Vector cascade algorithms with infinitely supported masks in weighted L 2-spaces

In this paper, we shall study the solutions of functional equations of the form Φ =∑α∈Zsa(α)Φ(M·-α), where Φ = (φ1, . . . , φr)T is an r×1 column vector of functions on the s-dimensional Euclidean space, a:=(a(α))α∈Zs is an exponentially decaying sequence of r×r complex matrices called refinement mask and M is an s×s integer matrix such that limn→∞M-n=0. We are interested in the question, for a mask a with exponential decay, if there exists a solution Φ to the functional equation with each function φj,j=1, . . . ,r, belonging to L2(Rs) and having exponential decay in some sense? Our approach will be to consider the convergence of vector cascade algorithms in weighted L2 spaces. The vector cascade operator Qa,M associated with mask a and matrix M is defined by Qa,Mf:=∑α∈Zsa(α)f (M·-α),f= (f1, . . . , fr)T∈(L2,μ(Rs))r.The iterative scheme (Qan,Mf)n=1,2,... is called a vector cascade algorithm or a vector subdivision scheme. The purpose of this paper is to provide some conditions for the vector cascade algorithm to converge in (L2,μ(Rs))r , the weighted L2 space. Inspired by some ideas in [Jia,R.Q.,Li,S.: Refinable functions with exponential decay: An approach via cascade algorithms. J. Fourier Anal. Appl., 17, 1008-1034 (2011)], we prove that if the vector cascade algorithm associated with a and M converges in (L2(Rs))r, then its limit function belongs to (L2,μ(Rs))r for some μ0.     

Converse theorems in the theory of approximate integration

We characterize certain function classes in terms of the remainder in the quadrature formula \(\int_{ - \infty }^\infty {f(x)dx = (\pi /\sigma )\sum\nolimits_{v = - \infty }^\infty {f(v\pi /\sigma ) + R_\sigma [f]} } \) . In the process, we prove a generalization of the famous theorem of Paley and Wiener about entire functions of exponential type belonging toL ² on the real line.     

Best constants of harmonic approximation of some convolution classes associated with Laplace operator in d-variables

In this paper,we consider the best EFET(entire functions of the exponential type) approximations of some convolution classes associated with Laplace operator on R d and obtain exact constants in the spaces L1(R2) and L2(Rd).Moreover,the best constants of trigonometric approximations of their analogies on Td are also gained.     

Spectral property of the Bernoulli convolutions

For 0<ρ<1, let μρ be the Bernoulli convolution associated with ρ. Jorgensen and Pedersen [P. Jorgensen, S. Pedersen, Dense analytic subspaces in fractal L²-spaces, J. Anal. Math. 75 (1998) 185-228] proved that if ρ=1/q where q is an even integer, then L²(μρ) has an exponential orthonormal basis. We show that for any 0<ρ<1, L²(μρ) contains an infinite orthonormal set of exponential functions if and only if ρ is the nth root of a fraction p/q where p is an odd integer and q is an even integer.     

О последовательност ях сумм Фурье-Уолша ограниченных функци й

Let L^∞ denote the space of measurable 1-periodic essentially bounded functionsf(x) with ∥f∥=vrai sup ¦f(x)¦,S _k (f, x) thek-th partial sum of the Walsh-Fourier series off(x),L _k thek-th Lebesgue constant. The following theorem is proved. Theorem. Letλ={λ _K } be a sequence of nonnegative numbers, $$\left\| \lambda \right\|_1 = \mathop \sum \limits_{k = 1}^\infty \lambda _k< \infty ,\left\| \lambda \right\|_2 = (\mathop \sum \limits_{k = 1}^\infty \lambda _k^2 )^{1/2} ,m = log[(\left\| \lambda \right\|_1 /\left\| \lambda \right\|_2 )]$$ .Then for an arbitrary function f∈L ^∞ the following inequalities hold true $$\begin{gathered} \left\| {\mathop \sum \limits_{k = 1}^\infty \lambda _k \left| {S_k (f,x)} \right|} \right\| \leqq \mathop \sum \limits_{k = 1}^\infty \lambda _k (L_{[k2 - 2m]} + c)\left\| f \right\|, \hfill \\ \hfill \\ \mathop \sum \limits_{k = 1}^\infty \lambda _k \left\| {S_k (f)} \right\| \leqq \mathop \sum \limits_{k = 1}^\infty \lambda _k (L_{[k2 - m]} + c)\left\| f \right\| \hfill \\ \end{gathered} $$ , where[y] denotes integral part of a number y>0 and c is an absolute constant. A corollary of the above theorem is that for each functionfεL ^∞ the Lebesgue estimate can be refined for a certain sequence of indices, while the growth order of Lebesgue constants along that sequence can be arbitrarily close to the logarithmic one. “In the mean”, however, the Lebesgue estimate is exact. A further corollary deals with strong summability.     

A characterization of multidimensional multi-knot piecewise linear spectral sequence and its applications

We characterize a class of piecewise linear spectral sequences. Associated with the spectral sequence, we construct an orthonormal exponential bases for L2（[0,1）d）, which are called generalized Fourier bases. Moreover, we investigate the convergence of Bochner-Riesz means of the generalized Fourier series.     

Series with monotone coefficients by Walsh system

The paper studies the series $\sum\limits_{n = 0}^\infty {a_n } W_n (x)$ by Walsh system, where |a _n | monotone tends to zero and $\sum\limits_{n = 1}^\infty {a_{_n }^2 } = \infty $ . Some theorems on correction in L ¹ and representability of functions from L ^p , p ∈ (0, 1) by subseries of the Walsh series are proved.     

Stabilization of the Korteweg-de Vries-Burgers equation with non-periodic boundary feedbacks

We consider the Korteweg-de Vries-Burgers (KdVB) equation on the domain [0,1]. We derive a control law which guaranteesL ²-global exponential stability,H ³-global asymptotic stability, andH ³-semiglobal exponential stability.     

On the Parameter-Uniform Convergence of Exponential Spline Interpolation in the Presence of a Boundary Layer

The paper is concerned with the problem of generalized spline interpolation of functions having large-gradient regions. Splines of the class C², represented on each interval of the grid by the sum of a second-degree polynomial and a boundary layer function, are considered. The existence and uniqueness of the interpolation L-spline are proven, and asymptotically exact two-sided error estimates for the class of functions with an exponential boundary layer are obtained. It is established that the cubic and parabolic interpolation splines are limiting for the solution of the given problem. The results of numerical experiments are presented.     

Analysis of a two-phase cell division model

In this paper we mainly study an equal mitosis two-phase cell division model. By using the C₀-semigroup theory, we prove that this model is the well-posed in L¹[0, 1] × L¹[0, 1], and exhibits asynchronous exponential growth phenomenon as time goes to infinity. We also give a comparison between this two-phase model with the classical one-phase model. Finally, we briefly study the asymmetric two-phase cell division model, and show that similar results hold for it.     

Poincaré type inequalities on the discrete cube and in the CAR algebra

We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {?1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and Götze. We also get inequalities between the L ^p norms of $ \left\vert \nabla f\right\vert We prove L ^p Poincaré inequalities with suitable dimension free constants for functions on the discrete cube {−1, 1} ⁿ . As well known, such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by Bobkov and G?tze. We also get inequalities between the L ^p norms of and moreover L ^p spaces may be replaced by more general ones. Similar results hold true, replacing functions on the cube by matrices in the *-algebra spanned by n fermions and the L ^p norm by the Schatten norm C _p .     

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 the spectra and eigenfunctions of the Schrödinger and Maxwell operators

The new sufficient conditions of the exponential decay of eigenfunctions and the absence of eigenvalues are obtained for the second-order elliptic operator in L²(rⁿ), generalizing the Schrödinger operator, and for the Maxwell operator with the matrix medium characteristics (the crystal optics operator) in L²(rⁿ).     

Heat kernel bounds for elliptic partial differential operators in divergence form with Robin-type boundary conditions

One of the principal topics of this paper concerns the realization of self-adjoint operators L _Θ,Ω in L ²(Ω; d ⁿ x) ^m , m, n ∈ ?, associated with divergence form elliptic partial differential expressions L with (nonlocal) Robin-type boundary conditions in bounded Lipschitz domains Ω ? ? ⁿ . In particular, we develop the theory in the vector-valued case and hence focus on matrix-valued differential expressions L which act as $$Lu = - \left( {\sum\limits_{j,k = 1}^n {\partial _j } \left( {\sum\limits_{\beta = 1}^m {a_{j,k}^{\alpha ,\beta } \partial _k u_\beta } } \right)} \right)_{1 \leqslant \alpha \leqslant m} , u = \left( {u_1 , \ldots ,u_m } \right).$$ The (nonlocal) Robin-type boundary conditions are then of the form $$v \cdot ADu + \Theta [u|_{\partial \Omega } ] = 0{\text{ on }}\partial \Omega ,$$ where Θ represents an appropriate operator acting on Sobolev spaces associated with the boundary ?Ω of Ω, ν denotes the outward pointing normal unit vector on ?Ω, and $Du: = \left( {\partial _j u_\alpha } \right)_{_{1 \leqslant j \leqslant n}^{1 \leqslant \alpha \leqslant m} } .$ Assuming Θ ≥ 0 in the scalar case m = 1, we prove Gaussian heat kernel bounds for L _Θ,Ω, by employing positivity preserving arguments for the associated semigroups and reducing the problem to the corresponding Gaussian heat kernel bounds for the case of Neumann boundary conditions on ?Ω. We also discuss additional zero-order potential coefficients V and hence operators corresponding to the form sum L _Θ,Ω + V.     

On first passage time structure of random walks

For continuous time birth-death processes on {0,1,2,…}, the first passage time T⁺_n from n to n + 1 is always a mixture of (n + 1) independent exponential random variables. Furthermore, the first passage time T_0,n+1 from 0 to (n + 1) is always a sum of (n + 1) independent exponential random variables. The discrete time analogue, however, does not necessarily hold in spite of structural similarities. In this paper, some necessary and sufficient conditions are established under which T⁺_n and T_0,n+1 for discrete time birth-death chains become a mixture and a sum, respectively, of (n + 1) independent geometric random variables on {1,2,…};. The results are further extended to conditional first passage times.