The error term in Nevanlinna’s inequality

An upper bound is given for the error termS(r, |a _j |,f) in Nevanlinna’s inequality. For given positive increasing functions p and $ with ∫ ₁ ^∞ dr/p(r) = ∫ ₁ ^∞ dr/r ?(r) = ∞, setP(r) = ∫ ₁ ^r dt/p,Ψ(r) = ∫ ₁ ^r dt/t ?(t) We prove that $$S(r, \left\{ {a_j } \right\}, f) \leqslant \log \frac{{T(r, f)\phi (T(r, f))}}{{p(r)}} + O(1)$$ holds, with a small exceptional set of r, for any finite set of points |a _j | in the extended plane and any meromorphic function f such thatΨ(T(r, f)) =O(P(r)). This improves the known results of A. Hinkkanen and Y. F. Wang. The sharpness of the estimate is also considered.     

Long time small solutions to nonlinear parabolic equations

A sharp result on global small solutions to the Cauchy problem $$u_t = \Delta u + f\left( {u,Du,D^2 u,u_t } \right)\left( {t > 0} \right),u\left( 0 \right) = u_0 $$ In Rⁿ is obtained under the the assumption thatf is C^1+r forr>2/n and ‖u ₀‖C²(R ⁿ ) +‖u ₀‖W ₁ ² (R ⁿ ) is small. This implies that the assumption thatf is smooth and ‖u ₀ ‖W ₁ ^k (R ⁿ )+‖u ₀‖W ₂ ^k (R ⁿ ) is small fork large enough, made in earlier work, is unnecessary.     

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.     

Jackson-type inequalities and widths of function classes in L 2

The sharp Jackson-type inequalities obtained by Taikov in the space L ₂ and containing the best approximation and the modulus of continuity of first order are generalized to moduli of continuity of kth order (k = 2, 3, ... ). We also obtain exact values of the n-widths of the function classes F(k, r, Φ) and F _k ^r (h), which are a generalization of the classes F(1, r, Φ) and F _k ^r (h) studied by Taikov.     

On the equality of Kolmogorov and relative widths of classes of differentiable functions

We obtain sharper estimates of the remainders in the expression for the least value of the multiplier M for which the Kolmogorov widths d _n (W _C ^r , C) and the relative widths K _n (W _C ^r ,MW _C ^j ,C) of the class W _C ^r with respect to the class MW _C ^j , j < r, where r ? j is odd, are equal.     

On the growth order of the rectangular partial sums of multiple non-orthogonal series

Пустьd-натуральное ч исло,Z ^d — множество на боров k=(k ₁, ...,k _d ), состоящих из неотрицательных цел ыхk _j ,Z ₊ ^d =k∈Z ^d :k≧1. Предположи м, что системаf _k (x):k∈Z ₊ ^d ? ?L₂(X,A, μ) и последовател ьностьa _k :k∈Z ₊ ^d . таковы, чт о для всех b∈Z^d и m∈Z ₊ ^d выполн ены неравенства (2) $$\left\| {\sum\limits_{b + 1 \leqq k \leqq b + m} {a_k f_k (x)} } \right\|_2^2 \leqq w^2 (m)\sum\limits_{b + 1 \leqq k \leqq b + m} {a_k^2 } $$ где последовательно сть {w(m): m∈Z ₊ ^d положительн а и не убывает. Например, есл иf _k (х) — квазистационарная система, то для соотве тствующей последовательности {ω(m) (2) имeeт Меcтo ДЛЯ ЛЮбОЙ ПОС ЛеДОВатеЛЬНОСТИ {a_k}. В работе получены оце нки порядка роста пря моугольных частных суммS _m (x)= =∑ a_kf_k(x) при maxmj→∞ как в случ ае {a_k}∈l₂, таки для {a_k}l₂. Эти оценки явля1≦k≦m 1≦j≦d ются новыми даже для о ртогональных кратны х рядов. Показано, что упомяну тые оценки в общем слу чае являются точными.     

A new upper bound for the complex Grothendieck constant

Let ? denote the real function $$\varphi (k) = k\smallint _0^{\pi /2} \frac{{cos^2 t}}{{\sqrt {1 - k^2 sin ^2 t} }}dt, - 1 \leqq k \leqq 1$$ and letK _G ^C be the complex Grothendieck constant. It is proved thatK _G ^C ≦8/π(k ₀+1), wherek ₀ is the (unique) solution to the equation?(k)=1/8π(k+1) in the interval [0,1]. One has 8/π(k ₀+1) ≈ 1.40491. The previously known upper bound isK _G ^C ≦e ^1?y ≈ 1.52621 obtained by Pisier in 1976.     

Multipliers for the Weyl transform and Laguerre expansions

Let P_k denote the projection of L²(R ^R ) onto the kth eigenspace of the operator (-δ+?x?² andS _N ^α =(1/A _N ^α )Σ _k ^N =0A _N?k ^α P _k . We study the multiplier transformT _N ^α for the Weyl transform W defined byW(T _N ^αf )=S _n ^αW(f) . Applications to Laguerre expansions are given.     

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 possible Turán densities

The Turán density π(F) of a family F of k-graphs is the limit as n → ∞ of the maximum edge density of an F-free k-graph on n vertices. Let Π _∞ ^(k) consist of all possible Turán densities and let Π _fin ^(k) ? Π _∞ ^(k) be the set of Turán densities of finite k-graph families. Here we prove that Π _fin ^(k) contains every density obtained from an arbitrary finite construction by optimally blowing it up and using recursion inside the specified set of parts. As an application, we show that Π _fin ^(k) contains an irrational number for each k ≥ 3. Also, we show that Π _∞ ^(k) has cardinality of the continuum. In particular, Π _∞ ^(k) ≠ Π _fin ^(k) .     

Singular integrals along Lipschitz curves with holomorphic kernels

If γ(x)=x+iA(x),tan ^?1‖A′‖_∞<ω<π/2,S _ω ⁰ ={z∈C}| |argz|<ω, or, |arg(-z)|<ω} We have proved that if φ is a holomorphic function in S _ω ⁰ and \(\left| {\varphi (z)} \right| \leqslant \frac{C}{{\left| z \right|}}\) , denotingT f (z)= ∫?(z-ζ)f(ζ)dζ, ?f∈C ₀(γ), ?z∈suppf, where C_c(γ) denotes the class of continuous functions with compact supports, then the following two conditions are equivalent:

1. T can be extended to be a bounded operator on L²(γ);
2. there exists a function ?₁ ∈H ^∞(S _ω ⁰ ) such that ?′₁(z)=?(z)+?(-z), ?z∈S _ω ⁰ ?z∈S _w ⁰ .

     

The dimenstion and bases of divergence-free splines; a homological approach

For a triangulated 2-dimensional region in \(\mathbb{R}^2 \) , let S′_m(Δ) be the vector space of all C′ functions F on Δ such that for any simplex σ∈Δ, F|σ is a polynomial of degree at most m. Let D′_m(Δ) be the vector space consisting of all pairs (F₁, F₂) with F_i∈S′_m(Δ), such that Σ_t(?F_t/?x_t=0, i.e., the pair is divergence-free. Both S′_m(Δ) and D′_m(Δ) can be described in terms of chain complexes using the usual boundary map of homology, and these complexes can be related by an epimorphism. When Δ is 2-disk the epimorphism gives the explicit result that \(\mathbb{R}^2 \) . Bases for D′_m(Δ) are derived from bases of S′ _m+1 ⁺¹ (Δ) via the epimorphism in this case.     

Approximation operators and tauberian constants

The explicit expression of the smallest constantC satisfying $$\mathop {lim}\limits_{\lambda \to \infty } \left| {t_{n(\lambda )}^{(1)} - t_{m(\lambda )}^{(2)} } \right| \leqq C. \mathop {lim sup}\limits_{n \to \infty } \left| {d_n } \right|$$ for all sequences {s _n} satisfying lim sup _n→∞ |d _n| <∞, where {t _n ⁽¹⁾ }, {t _n ⁽²⁾ } are two generalised Hausdorff transforms of {s _n }, {d _n} is the generalised (C, α)-transform (0≦α≦1) of {λ _n a _n} andn(λ, m(λ) are suitably related, is obtained. These results are obtained by using new properties of positive approximation operators and generalised Bernstein approximation operators.     

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 chromatic numbers of spheres in ℝ n

Let χ(S _r ^n?1 )) be the minimum number of colours needed to colour the points of a sphere S _r ^n?1 of radius $r \geqslant \tfrac{1} {2}$ in ? ⁿ so that any two points at the distance 1 apart receive different colours. In 1981 P. Erd?s conjectured that χ(S _r ^n?1 )→∞ for all $r \geqslant \tfrac{1} {2}$ . This conjecture was proved in 1983 by L. Lovász who showed in [11] that χ(S _r ^n?1 ) ≥ n. In the same paper, Lovász claimed that if $r < \sqrt {\frac{n} {{2n + 2}}}$ , then χ(S _r ^n?1 ) ≤ n+1, and he conjectured that χ(S _r ^n?1 ) grows exponentially, provided $r \geqslant \sqrt {\frac{n} {{2n + 2}}}$ . In this paper, we show that Lovász’ claim is wrong and his conjecture is true: actually we prove that the quantity χ(S _r ^n?1 ) grows exponentially for any $r > \tfrac{1} {2}$ .     

Weighted norm inequalities for the vectorvalued Hardy-Littlewood maximal functions of one parameter rectangles

Let N denote the Hardy-Littlewood maximal operator for the familyR of one parameter rectangles. In this paper, we obtain that for 1 _w ^p (l^r) to L _W ^P (l^r) if and only if w ∈ A_P(R); for 1≤p<∞, N is bounded from L _W ^P (l^r) to weak L _W ^P (l^r) if and only if W ∈ A_P(R). Here we say W∈A_p (1), if $$\begin{gathered} \mathop {sup}\limits_{R \in R} \left( {\tfrac{1}{{|R|}}\smallint _r wdx} \right)\left( {\tfrac{1}{{|R|}}\smallint _R w^{ - 1/(p - 1)} dx} \right)^{p - 1}< \infty ,1< p< \infty , \hfill \\ (Nw)(x) \leqslant Cw(x)a.e.,p = 1 \hfill \\ \end{gathered} $$ ,     

О разложениях в ряд ы о бобщенно-абсолютно-м онотонных функций

In an earlier paper [1] the notion of the so-called 〈?, GL_J>-absolutely monotonie functions was introduced, where ?≧1, {λ_{k k=0} ^∞ is an arbitrary non-increasing sequence of positive numbers. It was found that the condition \(\sum\limits_{\lambda _{k > 0} } {\lambda _k^{ - 1} } = + \infty \) is necessary in order to have the series expansion for any function f(x)∈〈?, λ_j). HereL ^k/? f/(x) are special integro-differential operators of fractional order, is a system of functions associated with the Mittag-Leffler type functions \(E_\varrho (z;\mu ) = \sum\limits_{k = 0}^\infty {z^n /\Gamma (\mu + \kappa /\varrho )} \) and with the sequence {λ_k}. In the present paper it is proved (in particular, see Theorem 3.2) that the expansion (*) is valid almost everywhere on (0,l) if ∑ λ _k ^?1 =+∞. This result contains, as a special case (when ?=1 and λ_k=0,k≧0) the known theorem of S. N. Bernstein on absolutely monotonic functions.     

Asymptotic expansions of solutions of equations with a deviating argument in Banach spaces

For the equation $$Lu = \frac{1}{i}\frac{{du}}{{dt}}\sum\nolimits_{j = 0}^m {A_j u} (l - h_j^0 - h_j^1 (t)) = f(t),$$ whereh ₀ ^o =0,h ₀ ¹ =0 (t) ≡ 0,h _j ^o = const > 0,h ₁ ^j (t),j= 1, ...,m are nonnegative continuously differentiable functions in [0, ∞), Aj are bounded linear operators, under conditions on the resolvent and on the right hand sidef(t), we have obtained an asymptotic formula for any solution u(t) from L₂ in terms of the exponential solutions u_k(t), k=1, ..., n, of the equation $$\frac{1}{i}\frac{{du}}{{dt}} - A_0 u - \sum\nolimits_{j = 0}^m {A_j u} (t - h_j^0 ) = 0,$$ connected with the poles λ_k, k=1, ..., n, of the resolvent R_λ in a certain strip.