Pointwise estimates of polynomials orthogonal on a circle with respect to a weight not belonging to the spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript> (<Emphasis Type="Italic">r</Emphasis> &gt; 1)

Two-sided pointwise estimates are established for polynomials that are orthogonal on the circle |z| = 1 with respect to the weight ?(τ): = h(τ)|sin(τ/2)|^?1 g(|sin(τ/2)|) (τ ∈ ?), where g(t) is a concave modulus of continuity slowly changing at zero such that t ^?1 g(t) ∈ L ¹[0, 1] and h(τ) is a positive function from the class C _2π with a modulus of continuity satisfying the integral Dini condition. The obtained estimates are applied to find the order of the distance from the point t = 1 to the greatest zero of a polynomial orthogonal on the segment [?1, 1].     

Inequalities between best polynomial approximations and some smoothness characteristics in the space <Emphasis Type="Italic">L</Emphasis> <Subscript>2</Subscript> and widths of classes of functions

We obtain exact constants in Jackson-type inequalities for smoothness characteristics Λ_k(f), k ∈ N, defined by averaging the kth-order finite differences of functions f ∈ L₂. On the basis of this, for differentiable functions in the classes L₂^r, r ∈ N, we refine the constants in Jackson-type inequalities containing the kth-order modulus of continuity ω_k. For classes of functions defined by their smoothness characteristics Λ_k(f) and majorants Φ satisfying a number of conditions, we calculate the exact values of certain n-widths.     

On an approach to the analysis of asymptotic properties of solutions of first-order ordinary delay differential equations

For the first-order ordinary delay differential equation

$$u'(t) + p(t)u(r(t)) = 0,$$

where p ∈ L _loc(?₊; ?₊), τ ∈ C(?₊; ?₊), τ(t) ≤ t for t ∈ ?₊, lim_t→+∞ τ(t) = +∞, and ?₊:= [0, ∞), we obtain new criteria for the existence of sign-definite and oscillating solutions, thus generalizing some earlier-known results.

     

Global hypoellipticity for first-order operators on closed smooth manifolds

The main goal of this paper is to address global hypoellipticity issues for the class of first-order pseudo-differential operators L = D_t + C(t, x,D_x), where (t, x) ∈ T × M, T is the one-dimensional torus, M is a closed manifold, and C(t, x,D_x) is a first-order pseudo-differential operator on M, smoothly depending on the periodic variable t. In the case of separation of variables, when C(t, x,D_x) = a(t)p(x,D_x) + ib(t)q(x,D_x), we give necessary and sufficient conditions for the global hypoellipticity of L. In particular, we show that the famous (P) condition of Nirenberg-Treves is neither necessary nor sufficient to guarantee the global hypoellipticity of L.     

Equiconvergence of expansions in multiple Fourier series and in fourier integrals with “lacunary sequences of partial sums”

We investigate the equiconvergence on T^N = [?π, π)^N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ L_p(T^N) and g ∈ L_p(R^N), p > 1, N ≥ 3, g(x) = f(x) on T^N, in the case where the “partial sums” of these expansions, i.e., S_n(x; f) and J_α(x; g), respectively, have “numbers” n ∈ Z^N and α ∈ R^N (n_j = [α_j], j = 1,..., N, [t] is the integral part of t ∈ R¹) containing N ? 1 components which are elements of “lacunary sequences.”     

Output stabilization for a class of uncertain systems

The system

$$\frac{{dx}}{{dt}} = A\left( \cdot \right)x + B\left( \cdot \right)u,{\kern 1pt} \frac{{dy}}{{dt}} = A\left( \cdot \right)y + B\left( \cdot \right)u + D\left( {C*y - v} \right)$$

where v = C*x is an output, u = S*y is a control, A(·) ∈ R ^{n × n} , B(·) ∈ R ^{n × (n–p)}, C ∈ R ^{n × (n–p)}, and D ∈ R ^{n × (n–p)}, is considered. The elements α_ij(·) and β_ij(·) of the matrices A(·) and B(·) are arbitrary functionals satisfying the conditions

$$\mathop {\sup }\limits_{\left( \cdot \right)} |{\alpha _{ij}}\left( \cdot \right)| < \infty \left( {i,j \in 1,n} \right),\mathop {\sup }\limits_{\left( \cdot \right)} |{\beta _{ij}}\left( \cdot \right)| < \infty \left( {i \in 1,n,j \in 1,n - p} \right).$$

It is assumed that A(·) ∈ Z ₁ ∪ Z ₃ and A*(·) ∈ Z ₁ ∪ Z ₃, where Z ₁ is the class of matrices in which the first p elements of the kth superdiagonal are sign-definite and the elements above them are sufficiently small. The class Z ₃ differs from Z _t1 in that the elements between this superdiagonal and the (k + 1)th row are sufficiently small. If k > p, then the elements of the p × p square in the upper left corner of the matrix are sufficiently small as well. By using special quadratic Lyapunov functions, a matrix D for which y(t)–x(t) → 0 exponentially as t → ∞ is first found, and then a matrix S for which the vectors x(t) and y(t) have the same property is constructed.

     

Best quadrature formulas for evaluating line integrals of the first kind for certain classes of functions and curves determined by moduli of continuity

The extremal problem of minimizing the error of approximate evaluation of a line integral of the first kind is considered for certain classes of functions and spatial curves determined by moduli of continuity.It is proved that if the endpoints of the interval [0, L] (where L is the length of the curve along which the integration is performed) are not included in the set of nodes of a quadrature formula for evaluating the line integral of the first kind, then the best quadrature formula for the classes m^(p) _ρ of functions and \({H^{{\omega _1}, \ldots ,{\omega _m}}}\) of curves is the midpoint rectangle formula. If the extreme points x = 0 and x = L of the interval are included in the set of nodes of a quadrature formula for approximately evaluating the line integral (such formulas are said to be Markov-type), then, for these classes, the best formula is the trapezoidal rule. Sharp error estimates for all considered classed of functions and curves are calculated and a generalization to more general classes is given.     

Kazhdan-Lusztig cells in some weighted Coxeter groups

Let(W,S) be a Coxeter group with S = I■J such that J consists of all universal elements of S and that I generates a finite parabolic subgroup W_I of W with w_0 the longest element of W_I. We describe all the left cells and two-sided cells of the weighted Coxeter group(W,S,L) that have non-empty intersection with W_J,where the weight function L of(W, S) is in one of the following cases:(i) max{L(s) | s ∈J} min{L(t)|t∈I};(ii) min{L(s)|s ∈J} ≥L(w_0);(iii) there exists some t ∈ I satisfying L(t) L(s) for any s ∈I-{t} and L takes a constant value L_J on J with L_J in some subintervals of [1, L(w_0)-1]. The results in the case(iii) are obtained under a certain assumption on(W, W_I).     

On idempotent <Emphasis Type="Italic">τ</Emphasis>-measurable operators affiliated to a von Neumann algebra

Let τ be a faithful normal semifinite trace on a von Neumann algebra M, let p, 0 < p < ∞, be a number, and let L_p(M, τ) be the space of operators whose pth power is integrable (with respect to τ). Let P and Q be τ-measurable idempotents, and let A ≡ P ? Q. In this case, 1) if A ≥ 0, then A is a projection and QA = AQ = 0; 2) if P is quasinormal, then P is a projection; 3) if Q ∈ M and A ∈ L_p(M, τ), then A² ∈ L_p(M, τ). Let n be a positive integer, n > 2, and A = Aⁿ ∈ M. In this case, 1) if A ≠ 0, then the values of the nonincreasing rearrangement μ_t(A) belong to the set {0} ∪ [‖A^n?2‖^?1, ‖A‖] for all t > 0; 2) either μ_t(A) ≥ 1 for all t > 0 or there is a t₀ > 0 such that μ_t(A) = 0 for all t > t₀. For every τ-measurable idempotent Q, there is aunique rank projection P ∈ M with QP = P, PQ = Q, and PM = QM. There is a unique decomposition Q = P + Z, where Z² = 0, ZP = 0, and PZ = Z. Here, if Q ∈ L_p(M, τ), then P is integrable, and τ(Q) = τ(P) for p = 1. If A ∈ L₁(M, τ) and if A = A³ and A ? A² ∈ M, then τ(A) ∈ R.     

Solvability of Mixed Problems for the Klein–Gordon–Fock Equation in the Class <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> for <Emphasis Type="Italic">p</Emphasis> ≥ 1

We prove that the mixed problem for the Klein–Gordon–Fock equation u _tt (x, t) ? u _xx (x, t) + au(x, t) = 0, where a ≥ 0, in the rectangle Q _T = [0 ≤ x ≤ l] × [0 ≤ t ≤ T] with zero initial conditions and with the boundary conditions u(0, t) = μ(t) ∈ L _p [0, T ], u(l, t) = 0, has a unique generalized solution u(x, t) in the class L _p (Q _T ) for p ≥ 1. We construct the solution in explicit analytic form.     

On the total multiplicity of zeros of generalized polynomials related to nonoscillating trajectories

For a continuous curve L = {x: x = Z(t), t ∈ [a, b]} in R ⁿ , we study the number of zeros of the function l _h (t) = 〈h, Z(t)〉, where h ∈ R ⁿ . We introduce the notion of multiple zeros for such functions and study the possibility of estimating the total multiplicity of such zeros under the assumption that the system {z ₁(t), z ₂(t), …, z _n (t)} of coordinates of the function Z(t) is a Chebyshev system on [a, b].     

Interpolation of operators of weak type (ϕ, ϕ)

Considering the measurable and nonnegative functions ? on the half-axis [0, ∞) such that ?(0) = 0 and ?(t) → ∞ as t → ∞, we study the operators of weak type (?, ?) that map the classes of ?-Lebesgue integrable functions to the space of Lebesgue measurable real functions on ?ⁿ. We prove interpolation theorems for the subadditive operators of weak type (?₀, ?₀) bounded in L _∞(?ⁿ) and subadditive operators of weak types (?₀, ?₀) and (?₁, ?₁) in L _?(? ⁿ ) under some assumptions on the nonnegative and increasing functions ?(x) on [0, ∞). We also obtain some interpolation theorems for the linear operators of weak type (?₀, ?₀) bounded from L _∞(?ⁿ) to BMO(? ⁿ). For the restrictions of these operators to the set of characteristic functions of Lebesgue measurable sets, we establish some estimates for rearrangements of moduli of their values; deriving a consequence, we obtain a theorem on the boundedness of operators in rearrangement-invariant spaces.     

Convexity,moduli of smoothness and a Jackson-type inequality

For a Banach space B of functions which satisfies for some m>0

$ \max ({\|F+G\|}_B,{\|F-G\|}_B)\geqq ({\|F\|}^s_B+m{\|G\|}^s_B)^{1/s},\quad \forall \,F,G\in B $

(?)

a significant improvement for lower estimates of the moduli of smoothness ω ^r (f,t) _B is achieved. As a result of these estimates, sharp Jackson inequalities which are superior to the classical Jackson type inequality are derived. Our investigation covers Banach spaces of functions on ? ^d or \(\mathbb{T}^{d}\) for which translations are isometries or on S ^d?1 for which rotations are isometries. Results for C ₀ semigroups of contractions are derived. As applications of the technique used in this paper, many new theorems are deduced. An L _p space with 1<p<∞ satisfies (?) where s=max??(p,2), and many Orlicz spaces are shown to satisfy (?) with appropriate s.

     

Integral averaging technique for oscillation of damped half-linear oscillators

This paper is concerned with the oscillatory behavior of the damped half-linear oscillator (a(t)?_p(x′))′ + b(t)?_p(x′) + c(t)?_p(x) = 0, where ?p(x) = |x|^p?1 sgn x for x ∈ ? and p > 1. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are oscillatory if and only if p ≠ 2 is presented.     

On decompositions of trigonometric polynomials

Let R _t [θ] be the ring generated over R by cosθ and sinθ, and R _t (θ) be its quotient field. In this paper we study the ways in which an element p of R _t [θ] can be decomposed into a composition of functions of the form p = R ? q, where R ∈ R(x) and q ∈ R _t (θ). In particular, we describe all possible solutions of the functional equation R ₁ ? q ₁ = R ₂ ? q ₂, where R ₁,R ₂ ∈ R[x] and q ₁, q ₂ ∈ R _t [θ].     

Estimates for sums of moduli of blocks in trigonometric Fourier series

We consider the following two problems. Problem 1: what conditions on a sequence of finite subsets A _k ? ? and a sequence of functions λ _k : A _k → ? provide the existence of a number C such that any function f ∈ L ₁ satisfies the inequality ‖U _A,Λ(f)‖ _p ≤ C‖f‖₁ and what is the exact constant in this inequality? Here, \(U_{\mathcal{A},\Lambda } \left( f \right)\left( x \right) = \sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {\lambda _k \left( m \right)c_m \left( f \right)e^{imx} } } \right|}\) and c _m (f) are Fourier coefficients of the function f ∈ L ₁. Problem 2: what conditions on a sequence of finite subsets A _k ? ? guarantee that the function \(\sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {c_m \left( h \right)e^{imx} } } \right|}\) belongs to L _p for every function h of bounded variation?     

Nonlinear trigonometric approximations of multivariate function classes

Order-sharp estimates are established for the best N-term approximations of functions from Nikol’skii–Besov type classes B_pq^sm(T^k) with respect to the multiple trigonometric system T^(k) in the metric of L_r(T^k) for a number of relations between the parameters s, p, q, r, and m (s = (s₁,..., s_n) ∈ R₊ⁿ, 1 ≤ p, q, r ≤ ∞, m = (m₁,..., m_n) ∈ Nⁿ, k = m₁ +... + m_n). Constructive methods of nonlinear trigonometric approximation—variants of the so-called greedy algorithms—are used in the proofs of upper estimates.     

Global weak solutions of the Cauchy problem for semilinear pseudo-hyperbolic equations

The Cauchy problem for a class of semilinear pseudo-hyperbolic equations is considered. For the corresponding linear problems, we obtain L _p → L _q estimates. By using these estimates, we prove global solvability theorems. We also establish the behavior of solutions as t → + ∞.     

Weighted Norm Inequalities for Spectral Multipliers on Graphs

Let Γ be a graph endowed with a reversible Markov kernel p, whose associated operator P is defined by \(Pf(x) = {\sum }_{y} p(x, y)f(y)\). We assume that the kernels p_n(x, y) associated to Pⁿ satisfy Gaussian upper bounds but do not assume they satisfy the Hölder continuity property and the temporal regularity. Denote by L = I ? P the discrete Laplacian on Γ. This article shows the weighted weak type (1, 1) estimates and the weighted L^p norm inequalities for the spectral multipliers of L. We also obtain the weighted L^p norm inequalities for the commutators of the spectral multipliers of L with BMO functions which are new even for the unweighted case.