Cantor-Lebesgue theorem for double trignometric series

Let ∥·∥ be a norm in R² and let γ be the unit sphere induced by this norm. We call a segment joining points x,y ε R² rational if (x₁ ? y₁)/(x₂ ? y₂) or (x₂ ? y₂)/(x₁ ? y₁) is a rational number. Let γ be a convex curve containing no rational segments. Satisfaction of the condition $$T_\nu (x) = \sum\nolimits_{\parallel n\parallel = \nu } {c_n e^{2\pi i(n_1 x_1 + n_2 x_2 )} } \to 0(\nu \to \infty )$$ in measure on the set e? [- 1/2,1/2)×[- 1/2, 1/2) =T² of positive planar measure implies ∥T _v ∥L₄ (T²) → 0(v → ∞). if, however, γ contains a rational segment, then there exist a sequence of polynomials {T _v } and a set E ? T², ¦E¦ > 0, such that T _v (x) → 0(v → ∞) on E; however, ¦c_n¦ ? 0 for ∥n∥ → ∞.     

A note on approximation to the identity with iterated kernels

Пустьl ₁ иl ₂ — неотрицательные убывающие функции на (0, ∞). Допустим, что $$\int\limits_0^\infty {S^{n_i - 1} l_i (S)\left( {1 + \log + \frac{1}{{S^{n_i } l_i (S)}}} \right)dS}< \infty ,$$ , гдеn ₁ иn ₂ — натуральные числа. Тогда для каждой функции \(f \in L^1 (R^{n_1 + n_2 } )\) при почти всех (x₀, у₀) мы имеем $$\mathop {\lim }\limits_{\lambda \to \infty } \lambda ^{n_1 + n_2 } \int\limits_{R^{n_1 } } {\int\limits_{R^{n_2 } } {l_1 } } (\lambda |x|)l_2 (\lambda |y|)f(x_0 - x,y_0 - y)dx dy = f(x_0 ,y_0 )\int\limits_{R^{n_1 } } {\int\limits_{R^{n_2 } } {l_i (|x|)l_2 } } (|y|)dx dy.$$      

An example of a second-order nonhypoelliptic operator with the property of global hypoellipticity

It is proved that the operator $$P \equiv - \frac{{\partial ^2 }}{{\partial x_1^2 }} - \sum\nolimits_{k = 2}^n {\frac{\partial }{{\partial x_k }}\varphi ^2 (x)} \frac{\partial }{{\partial x_k }},$$ where ? ε C^∞(Ω) (Ω is a domain in Rⁿ), {x: ?(x) = 0} is a compactun in Ω which is the closure of its internal points, has the property of global hypoellipticity in Ω, i.e., $$\begin{array}{*{20}c} {v \in D'(\Omega ),} & {Pv \in C^\infty } & {(\Omega ) \Rightarrow \upsilon \in C^\infty (\Omega ).} \\ \end{array} $$ . This operator is not hypoelliptic.     

Kolmogorov-type inequalities and the best formulas for numerical differentiation

For a certain class of complex-valued functionsf(x), ?∞ $$u_N = \mathop {\inf }\limits_{\parallel A\parallel \leqslant N_\parallel f^{(n)} \parallel _{L_2 \leqslant } 1} \parallel f^{(k)} - A(f)\parallel C$$ of a differential operator by linear operators A with the norm ∥A∥ _L2 ^C ≤N,N,>0. Using the value u_N, the smallest constant Q in the inequality $$\parallel f^{(k)} \parallel _Q \leqslant Q\parallel f\parallel _{L_2 }^\alpha \parallel f^{(n)} \parallel _{L_2 }^\beta $$ is found.     

Asymptotic formula for the mean value of a multiple trigonometric sum

When k≥k₀=10 Mr²n log (rn) we have for the trigonometric integral $$J_n (k,P) = \int_E {|S(A)|^{2k} dA,} $$ where $$\begin{gathered} S(A) = \sum _{x_1 = 1}^P \cdots \sum _{x_r = 1}^P \exp (2\pi if_A (x_1 , \ldots ,x_r )), \hfill \\ f_A (x_1 , \ldots ,x_r ) = \sum _{t_1 = 0}^n \cdots \sum _{t_r = 0}^n \alpha _{t_1 \cdots l_r } x_1^{t_1 } \cdots x_{r^r }^t \hfill \\ \end{gathered} $$ and E is the M-dimensional unit cube, the asymptotic formula $$J_n (k,P) = \sigma \theta P^{2kr - rnM/2} + O(P^{2kr - rnM/2 - 1/(2M)} ) + O(P^{2kr - rnM/2 - 1/(500r^2 \log (rn))} ),$$ where σ is a singular series and θ is a singular integral.     

Best quadrature formula on the class W * r L2 of periodic functions

For the classes of periodic functions with r-th derivative integrable in the mean,we obtain a best quadrature formula of the form $$\begin{gathered} \int_0^1 {f(x)dx = \sum\nolimits_{k = 0}^{m - 1} {\sum\nolimits_{l = 0}^\rho {p_{k,l} } } } f^{(l)} (x_k ) + R(f),0 \leqslant \rho \leqslant r - 1, \hfill \\ 0 \leqslant x_0< x_1< ...< x_{m - 1} \leqslant 1, \hfill \\ \end{gathered}$$ where ρ=r?2 and r?3, r=3, 5, 7, ..., and we determine an exact bound for the error of this formula.     

Constants related to operators of class Cρ

Let T be a Hilbert-space operator of class $$\left\| {S^{ - 1} TS} \right\| \leqslant 1\,\,\,\,and\,\,\left\| {S^{ - 1} } \right\| \cdot \left\| S \right\| \leqslant \max (1,\rho )$$ in the sense of Sz.-Nagy and Foia?. Then there exists an invertible operator S such that ∥S^?1TS∥≤1 and ∥S^?1∥·∥S∥≤max(1,ρ). The following estimate is valid for the bilateral limit: $$\mathop {lim}\limits_{n \to \infty } \{ \left\| {T^n h} \right\|^2 + \left\| {T*^n h} \right\|^2 \} \leqslant max(2,\rho )\left\| h \right\|^2 .$$ Here the constants max(1,ρ) and max(2,ρ) are the best possible in their respective cases.     

Estimates for the coefficients of univalent functions in terms of the second coefficient

For the coefficients b_n of an odd function \(f(z) = z + \sum\nolimits_{k = 1}^\infty {{}^bk^{z^{2k + 1} } } \) , regular in the unit disk, we obtain the estimate $$|b_n | \leqslant \frac{1}{{\sqrt 2 }}\sqrt {1 + |b_1 |^2 } \exp \frac{1}{2}\left( {\delta + \frac{1}{2}|b_1 |^2 } \right),where \delta = 0.312,$$ (1) from which it follows that ¦b_n¦≤1, if ¦b₁¦≤0.524. It follows from (1) that the coefficients c_n, n = 3, 4,..., of a regular function \(f(2) = z + \sum\nolimits_{k = 2}^\infty {{}^ck^{z^k } } \) , univalent in the unit desk, satisfy $$|c_n | \leqslant \frac{1}{2}\left( {1 + \frac{{|c_2 |^2 }}{4}} \right)n\exp \left( {\delta + \frac{{|c_2 |^2 }}{8}} \right),where \delta = 0.312,$$ in particular, ¦c_n¦≤n, if ¦c₂¦≤1.046.     

A mixed boundary-value problem for a hyperbolic-parabolic equation

Let Ω be a bounded domain in the n-dimensional Euclidean space. In the cylindrical domain QT=Ω x [0, T] we consider a hyperbolic-parabolic equation of the form (1) $$Lu = k(x,t)u_{tt} + \sum\nolimits_{i = 1}^n {a_i u_{tx_i } - } \sum\nolimits_{i,j = 1}^n {\tfrac{\partial }{{\partial x_i }}} (a_{ij} (x,t)u_{x_j } ) + \sum\nolimits_{i = 1}^n {t_i u_{x_i } + au_t + cu = f(x,t),} $$ where \(k(x,t) \geqslant 0,a_{ij} = a_{ji} ,\nu |\xi |^2 \leqslant a_{ij} \xi _i \xi _j \leqslant u|\xi |^2 ,\forall \xi \in R^n ,\nu > 0\) . The classical and the “modified” mixed boundary-value problems for Eq. (1) are studied. Under certain conditions on the coefficients of the equation it is proved that these problems have unique solution in the Sobolev spaces W ₂ ¹ (QT) and W ₂ ² (QT).     

On absolute summation of Fourier series by subsequences

В работе изучается сл едующая задача. Пусть заданы числа 0<α≦1 и β<α. При каки х условиях на строго во зрастающую последов ательность натуральных чисел {n _k } _k ^t8 =1 для всех 2π-периодических функ ций \(f(x) \sim \sum\limits_{v = - \infty }^\infty {c_v e^{ivx} } \) , принадлежащих к лассу Lip α, равномерно пох будет выполнено неравенство $$\sum\limits_{k = 1}^\infty {|\sum\limits_{n_k \leqq |v|< n_{k + 1} } {c_v e^{ivx} } |n_k^\beta< \infty ?} $$ .     

On the rational (K + 1,K + 1)-type difference equation

In this paper we investigate the boundedness character of the positive solutions of the rational difference equation of the form $$x_{n + 1} = \frac{{a_0 + \sum\nolimits_{j = 1}^k {a_j x_{n - j + 1} } }}{{b_0 + \sum\nolimits_{j = 1}^k {b_j x_{n - j + 1} } }}, n = 0,1,...$$ where k ε N, andaj,bj, j = 0,1,…, k, are nonnegative numbers such thatb ₀+∑ _j=1 ^k b _j x _n-j+1>0 for everyn ∈N ∪{0}. In passing we confirm several conjectures recently posed in the paper: E. Camouzis, G. Ladas and E. P. Quinn, On third order rational difference equations (part 6).     

Cones and error bounds for linear iterations

Consider an ordered Banach space with a cone of positive elementsK and a norm ∥·∥. Let [a,b] denote an order-interval; under mild conditions, ifx*∈[a,b] then $$||x * - \tfrac{1}{2}(a + b)|| \leqslant \tfrac{1}{2}||b - a||.$$ This inequality is used to generate error bounds in norm, which provide on-line exit criteria, for iterations of the type $$x_r = Ax_{r - 1} + a,A = A^ + + A^ - ,$$ whereA ⁺ andA ^? are bounded linear operators, withA ⁺ K ?K andA ^? K ? ?K. Under certain conditions, the error bounds have the form $$\begin{gathered} ||x * - x_r || \leqslant ||y_r ||,y_r = (A^ + - A^ - )y_{r - 1} , \hfill \\ ||x * - x_r || \leqslant \alpha ||\nabla x_r ||, \hfill \\ ||x * - \tfrac{1}{2}(x_r + x_{r - 1} )|| \leqslant \tfrac{1}{2}||\nabla x_r ||. \hfill \\ \end{gathered} $$ These bounds can be used on iterative methods which result from proper splittings of rectangular matrices. Specific applications with respect to certain polyhedral cones are given to the classical Jacobi and Gauss-Seidel splittings.     

The C-convexity of Banach spaces with unconditional bases

A Banach space is called C-convex if the space c₀ cannot be represented finitely in it. Necessary and sufficient conditions for the C-convexity of a space with an unconditional basis and of the product of a space Y with respect to the unconditional basis of a space X are obtained. These conditions are rendered concrete for two classes of spaces: The Orlich space of sequences is C-convex if and only if its normalizing function satisfies the δ₂-condition; the Lorentz space of sequences is C-convex if and only if its normalizing sequence satisfies the condition \(\mathop {\underline {\lim } }\limits_{n \to \infty } {{\sum\nolimits_{i = 1}^{2n} {c_i } } \mathord{\left/ {\vphantom {{\sum\nolimits_{i = 1}^{2n} {c_i } } {\sum\nolimits_{i = 1}^n {c_i > 1} }}} \right. \kern-0em} {\sum\nolimits_{i = 1}^n {c_i > 1} }}\) . We call a Banach space X a C-convex space if the following condition is fulfilled: $$\mathop {\sup }\limits_n \inf d\left( {X_n , l_\infty ^n } \right) = \infty $$ ,     

Separate asymptotics of two series of eigenvalues for a single elliptic boundary-value problem

The spectral problem in a bounded domain Ω?Rⁿ is considered for the equation Δu= λu in Ω, ?u=λ?υ/?ν on the boundary of Ω (ν the interior normal to the boundary, Δ, the Laplace operator). It is proved that for the operator generated by this problem, the spectrum is discrete and consists of two series of eigenvalues {λ _j ⁰ } _j=1 ^∞ and {λ _j ^∞ } _j=1 ^∞ , converging respectively to 0 and +∞. It is also established that $$N^0 (\lambda ) = \sum\nolimits_{\operatorname{Re} \lambda _j^0 \geqslant 1/\lambda } {1 \approx const} \lambda ^{n - 1} , N^\infty (\lambda ) \equiv \sum\nolimits_{\operatorname{Re} \lambda _j^\infty \leqslant \lambda } {1 \approx const} \lambda ^{n/1} .$$ The constants are explicitly calculated.     

Estimate of a multidimensional sum with the Legendre symbol for a polynomial of odd degree

The estimate $\left| {\sum\nolimits_{x_1 ,...,x_n \in F_q } {x(f(x_1 ,...,x_n ))} } \right| \leqslant (d - 1)^n q^{n/2} $ is derived for the quadratic character Λ of a field F_q of q elements and a polynomial f of odd degree d over F_q under certain natural conditions.     

О коэффициентах рядо в Фурье по множествам

The following statement is proved: Theorem.Let f(x), 0≦x≦2π, possess the Fourier expansion $$\mathop \sum \limits_{\kappa = - \infty }^\infty c_\kappa e^{in} \kappa ^x with \bar c_\kappa = c_{ - \kappa } , n_\kappa = - \bar n_{ - \kappa }$$ where {n _k } is a Sidon sequence. Then in order to have $$\mathop \sum \limits_{\kappa = - \infty }^\infty |c_\kappa |^p< \infty$$ for a given p, 1 $$\mathop \sum \limits_{k = 1}^\infty \left( {\frac{{\left\| f \right\|L^k (0,2\pi )}}{k}} \right)^p< \infty$$ . An analogous statement holds true for series with respect to the Rademacher system.     

The absolute convergence of lacunary series

A theorem is proved from which it follows that there exists a complete U-set E and a number p such that: a) if the p-lacunary trigonometric series $$\sum\nolimits_{k = 1}^\infty {a_k \sin (n_k x + \varepsilon _k ),} \frac{{\lim }}{{k \to \infty }}n_{k + 1} /n_k > p,$$ converges on E, the series of the moduli of its coefficients converges; b) if the sum of the p-lacunary trigonometric series is differentiable on E, it is continuously differentiable everywhere.     

Convergence of Riemann sums for functions which can be represented by trigonometric series with coefficients forming a monotonic sequence

The following theorem is proved. If $$f(x) = \frac{{\alpha _0 }}{2} + \sum\nolimits_k^\infty \alpha _k \cos 2\pi kx + b_k \sin 2\pi kx,$$ wherea _k ↓ 0 and b_k ↓ 0, then $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\nolimits_{s = 0}^{n - 1} {f\left( {x + \frac{s}{n}} \right) = \frac{{\alpha _0 }}{2}} $$ on (0, 1) in the sense of convergence in measure. If in additionf(x) ε L² (0, 1), then this relation holds for almost all x.     

On cubic polynomials II. Multiple exponential sums

Letf(x ₁,...,x _s ) be a cubic polynomial with integer coefficients,q a prime power, ande(z)=e ^2πiz . We are going to estimate sums $$\sum\limits_{x_1 = 1}^q { \cdot \cdot \cdot } \sum\limits_{x_S = 1}^q {e(q^{ - 1} f(x_1 ,...,x_S ))} $$ , as well as generalizations of such sums.