Convergence of polyharmonic splines on semi-regular grids \mathbb{Z}{{\boldsymbol{\times}} \boldsymbol{a}} {\mathbb{Z}^{\it\boldsymbol{n}}} for {\boldsymbol{a}\rightarrow\mathbf {0}}

Let p, n ∈ ? with 2p ≥ n + 2, and let I _a be a polyharmonic spline of order p on the grid ? × a? ⁿ which satisfies the interpolating conditions $I_{a}\left( j,am\right) =d_{j}\left( am\right) $ for j ∈ ?, m ∈ ? ⁿ where the functions d _j : ? ⁿ → ? and the parameter a > 0 are given. Let $B_{s}\left( \mathbb{R}^{n}\right) $ be the set of all integrable functions f : ? ⁿ → ? such that the integral $$ \left\| f\right\| _{s}:=\int_{\mathbb{R}^{n}}\left| \widehat{f}\left( \xi\right) \right| \left( 1+\left| \xi\right| ^{s}\right) d\xi $$ is finite. The main result states that for given $\mathbb{\sigma}\geq0$ there exists a constant c>0 such that whenever $d_{j}\in B_{2p}\left( \mathbb{R}^{n}\right) \cap C\left( \mathbb{R}^{n}\right) ,$ j ∈ ?, satisfy $\left\| d_{j}\right\| _{2p}\leq D\cdot\left( 1+\left| j\right| ^{\mathbb{\sigma}}\right) $ for all j ∈ ? there exists a polyspline S : ? ⁿ⁺¹ → ? of order p on strips such that $$ \left| S\left( t,y\right) -I_{a}\left( t,y\right) \right| \leq a^{2p-1}c\cdot D\cdot\left( 1+\left| t\right| ^{\mathbb{\sigma}}\right) $$ for all y ∈ ? ⁿ , t ∈ ? and all 0 < a ≤ 1.     

Modal stabilization of a certain class of uniformly controllable systems

The system $\dot x = A( \cdot )x + b( \cdot )u,$ where A(·) ∈ ? ^n×n and b(·) ∈ ? ^n×1, is considered. The elements of the matrix A(·) and the column b(·) are bounded by nonanticipating functionals of an arbitrary nature that satisfy the condition $\mathop {\inf }\limits_{( \cdot )} A^{n - 1} ( \cdot )b( \cdot ),...,A( \cdot )b( \cdot ),b( \cdot )| > 0$ . From a given constant spectrum contained in the left half-plane, a feedback u = (s(·), x) is constructed, the coefficients of which are expressed in terms of A(·) and b(·). Conditions for the closed system to be globally exponentially stable are found. A similar result is obtained for the system $x(k + 1) = A(k)x(k) + b(k)u(k)$ .     

On oscillatory integral Hilbert transformation with trigonometric polynomial phase

Let $ \mathcal{P}_n $ denote the set of algebraic polynomials of degree n with the real coefficients. Stein and Wpainger [1] proved that $$ \mathop {\sup }\limits_{p( \cdot ) \in \mathcal{P}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{ip(x)} }} {x}dx} } \right| \leqslant C_n , $$ where C _n depends only on n. Later A. Carbery, S. Wainger and J. Wright (according to a communication obtained from I. R. Parissis), and Parissis [3] obtained the following sharp order estimate $$ \mathop {\sup }\limits_{p( \cdot ) \in \mathcal{P}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{ip(x)} }} {x}dx} } \right| \sim \ln n. $$ . Now let $ \mathcal{T}_n $ denote the set of trigonometric polynomials $$ t(x) = \frac{{a_0 }} {2} + \sum\limits_{k = 1}^n {(a_k coskx + b_k sinkx)} $$ with real coefficients a _k , b _k . The main result of the paper is that $$ \mathop {\sup }\limits_{t( \cdot ) \in \mathcal{T}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{it(x)} }} {x}dx} } \right| \leqslant C_n , $$ with an effective bound on C _n . Besides, an analog of a lemma, due to I. M. Vinogradov, is established, concerning the estimate of the measure of the set, where a polynomial is small, via the coefficients of the polynomial.     

Reelle Abstandsräume und hyperbolische Geometrie

Ein reeller Abstandsraum ist eine Menge S ≠ ø zusammen mit einer Abbildung d: S × S → ?. Für x,y ∈ S hei\t d(x,y) der Abstand von x und y. Für beliebige reelle Abstandsräume definieren wir Begriffe wie Gerade, sphärischer Teilraum, konvexe Teilmenge, Winkelma\e usf. derart, da\ diese Begriffe im Falle $$={\rm R}^{n}\ {\rm und}\ d(x,y)=\sqrt{x^{2}+y^{2}- 2xy}$$ klassische Objekte gleichen Namens ergeben. Sodann wenden wir unseren Begriffsapparat auf $$ S={\rm R}^{n}\ {\rm und}\ d(x,y)=\sqrt{1+x^{2}}\sqrt{1+y^{2}}- xy $$ an. Dabei entsteht dann die n-dimensionale hyperbolische Geometrie mit dem ?ⁿ als Punktmenge und geeigneten Geraden, Strecken usf. als dortige Grundobjekte.     

Semigroups of Jensen set-valued functions

A set-valued function F defined on a convex cone S in a real vector space X into the set n(Y) of all non-empty subsets of a real vector space Y is said to be the Jensen function iff $${1\over 2}\lbrack F(x)+F(x)\rbrack=F\lbrack{1\over 2}(x+y)\rbrack$$ for all x, y ∈ S. This note deals with iteration semigroups of Jensen set-valued functions on S.     

Асимптотические рав енства для интеграло в от модуля суммы кратн ого ряда из синусов

Assume that the coefficients of the series $$\mathop \sum \limits_{k \in N^m } a_k \mathop \Pi \limits_{i = 1}^m \sin k_i x_i $$ satisfy the following conditions: a) a_k → 0 for k₁ + k₂ + ...+k_m →∞, b) \(\delta _{B,G}^M (a) = \mathop {\mathop \sum \limits_{k_i = 1}^\infty }\limits_{i \in B} \mathop {\mathop \sum \limits_{k_j = 2}^\infty }\limits_{j \in G} \mathop {\mathop \sum \limits_{k_v = 0}^\infty }\limits_{v \in M\backslash (B \cup G)} \mathop \Pi \limits_{i \in B} \frac{1}{{k_i }}|\mathop \sum \limits_{I_j = 1}^{[k_j /2]} (\nabla _{l_G }^G (\Delta _1^{M\backslash B} a_k ))\mathop \Pi \limits_{j \in G} l_j^{ - 1} |< \infty ,\) for ∨B?M, ∨G?M,B∩G, where M={1,2, ...,m}, $$\begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,\Delta _1^j a_k = a_k - a_{k_{M\backslash \{ j\} } ,k_{j + 1} } ,\Delta _1^B a_k = \Delta _1^{B\backslash \{ j\} } (\Delta _1^j a_k ), \hfill \\ \Delta _{l_j }^j a_k = a_{k_{M\backslash \{ j\} } ,k_j - l_j } - a_{k_{M\backslash \{ j\} } ,k_j + l_j } ,\nabla _{l_G }^G a_k = \nabla _{l_{G\backslash \{ j\} } }^{G\backslash \{ j\} } (\nabla _{l_j }^j a_k ). \hfill \\ \end{gathered} $$ Then for all n∈N^m the following asymptotic equation is valid: $$\mathop \smallint \limits_{{\rm T}_{\pi /(2n + 1)}^m } |\mathop \sum \limits_{k \in N^m } a_k \mathop \Pi \limits_{i \in M} \sin k_i x_i |dx = \mathop \sum \limits_{k = 1}^n \left| {a_k } \right|\mathop \Pi \limits_{i \in M} k^{ - 1} + O(\mathop {\mathop \sum \limits_{B,{\mathbf{ }}G \subset M} }\limits_{B \ne M} \delta _{B,G}^M (a)).$$ Here \(T_{\pi /(2n + 1)}^m = \left\{ {x = (x1,x2,...,xm):\pi /(2n + 1) \leqq xi \leqq \pi ;i = \overline {1,m} } \right\}\) . In the one-dimensional case such an equation was proved by S. A. Teljakovskii.     

Congruences for 7 and 49-regular partitions modulo powers of 7

Let \(b_{k}(n)\) denote the number of k-regular partitions of n. In this paper, we prove Ramanujan-type congruences modulo powers of 7 for \(b_{7}(n)\) and \(b_{49}(n)\). For example, for all \(j\ge 1\) and \(n\ge 0\), we prove that

$$\begin{aligned} b_{7}\Bigg (7^{2j-1}n+\frac{3\cdot 7^{2j-1}-1}{4}\Bigg )\equiv 0\pmod {7^{j}} \end{aligned}$$

and

$$\begin{aligned} b_{49}\Big (7^{j}n+7^{j}-2\Big )\equiv 0\pmod {7^{j}}. \end{aligned}$$

     

A Tauberian theorem for ℓ-adic sheaves on \mathbb{A} 1

Let K ∈ L ¹(?) and let f ∈ L ^∞(?) be two functions on ?. The convolution $$ \left( {K*F} \right)\left( x \right) = \int_\mathbb{R} {K\left( {x - y} \right)f\left( y \right)dy} $$ can be considered as an average of f with weight defined by K. Wiener’s Tauberian theorem says that under suitable conditions, if $$ \mathop {\lim }\limits_{x \to \infty } \left( {K*F} \right)\left( x \right) = \mathop {\lim }\limits_{x \to \infty } \int_\mathbb{R} {\left( {K*A} \right)\left( x \right)} $$ for some constant A, then $$ \mathop {\lim }\limits_{x \to \infty } f\left( x \right) = A $$ We prove the following ?-adic analogue of this theorem: Suppose K, F, G are perverse ?-adic sheaves on the affine line $ \mathbb{A} $ over an algebraically closed field of characteristic p (p ≠ ?). Under suitable conditions, if $ \left( {K*F} \right)|_{\eta _\infty } \cong \left( {K*G} \right)|_{\eta _\infty } $ , then $ F|_{\eta _\infty } \cong G|_{\eta _\infty } $ , where η _∞ is the spectrum of the local field of $ \mathbb{A} $ at ∞.     

Dissipativity of some class of uncertain systems

We consider the system $$ \dot x = A\left( \cdot \right)x + B\left( \cdot \right)u, u = S\left( \cdot \right)x, t \geqslant t_0 , $$ where A(·) ∈ ? ^n×n , B(·) ? ^n×p , and S(·) ∈ ? ^p×n . The entries of matrices A(·), B(·), and S(·) are arbitrary bounded functionals. We consider the problem of constructing a matrix H > 0 and finding relations between the entries of the matrices B(·) and S(·) such that for a given constant matrix R the inequality $$ V\left( {x\left( t \right)} \right) < V\left( {x\left( {t_0 } \right)} \right) + \int\limits_{t_0 }^t {x*\left( \tau \right)Rx\left( \tau \right)d\tau ,} $$ where V(x) = x*Hx, is satisfied. This problem is solved for the cases where matrix A(·) has p sign-definite entries on the upper part of some subdiagonal or on the lower part of some superdiagonal. It is assumed also that all entries located to the left (or to the right) of the sign-definite entries are equal to zero.     

On properties of integrals of the Legendre polynomial

Properties of the integrals $P_{n0} (x) = P_n (x),P_{nk} (x) = \int\limits_{ - 1}^x {P_{n,k - 1} (y)dy} $ of the Legendre polynomials P _n (x) on the base interval ?1 ≤ x ≤ 1 are systematically considered. The generating function $(1 - 2xz + z^2 )^{k - 1/2} = Q_k (x,z) + ( - 1)^k (2k - 1)!!\sum\limits_{n = k}^\infty {P_{nk} (x)z^{n + k} } $ is defined; here, Q ₀ = 0 and Q _k with k > 0 is a polynomial of degree 2k ? 1 in each of the variables x and z. A second-order differential equation is derived, an analogue of Rodrigues’ formula is obtained, and the asymptotic behavior as n → ∞ is determined. It is proved that the representation $P_{nk} (x) = (x^2 - 1)^k f_{nk} (x)$ holds if and only if n ≥ k, where f _nk is a polynomial divisible by neither x ? 1 nor x + 1. The main result is the sharp bound $|P_{nk} (\cos \theta )| < \frac{{A_k }} {{\nu ^{k + 1/2} }}\sin ^{k - 1/2} \theta ,n \geqslant k.$ Here, $\nu ^2 = \left( {n + \frac{1} {2}} \right)^2 - \left( {k^2 - \frac{1} {4}} \right)\left( {1 - \frac{4} {{\pi ^2 }}} \right),A_k = \sqrt t _k J_k (t_k ) \sim \mu _1 k^{1/6} ,\mu _1 = 0.674885, $ where t _k is the maximum of the function $\sqrt t J_k (t)$ on the half-axis t > 0 and J _k (t) is the Bessel function. The first values A _k and differences A _k ? μ₁ k ^1/6 are tabulated below as follows:      

Hadamard Inequality and a Refinement of Jensen Inequality for Set—Valued Functions

We present the following set-valued analogue of the Hadamard inequality: Let Y be a Banach space, I be an open interval and let F: I ? cl(Y) be a continuous and convex set-valued function. Then $${F(a)+F(b)\over 2}\subset{1\over b-a}\int_a^b\ F(x)dx\subset F \bigg({a+b\over 2}\bigg)$$ , for every a, b ∈ I, a < b. Some refinement of Jensen inequality for set-valued functions is also given.     

Pointwise estimates for Lagrange interpolation polynomials

Let f ∈ C[?1, 1]. Let the approximation rate of Lagrange interpolation polynomial of f based on the nodes $ \left\{ {\cos \frac{{2k - 1}} {{2n}}\pi } \right\} \cup \{ - 1,1\} $ be Δ _{n + 2}(f, x). In this paper we study the estimate of Δ _{n + 2}(f,x), that keeps the interpolation property. As a result we prove that $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left| {T_n (x)} \right|\ln (n + 1) + \omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}\left| {T_n (x)} \right|} \right)} \right\}, $$ where T _n (x) = cos (n arccos x) is the Chebeyshev polynomial of first kind. Also, if f ∈ C ^r [?1, 1] with r ≧ 1, then $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\frac{{\sqrt {1 - x^2 } }} {{n^r }}\left| {T_n (x)} \right|\omega \left( {f^{(r)} ,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left( {\left( {\sqrt {1 - x^2 } + \frac{1} {n}} \right)^{r - 1} \ln (n + 1) + 1} \right)} \right\}. $$      

On a result of Hardy and Littlewood concerning Fourier series

Let f ∈ L ¹( $ \mathbb{T} $ ) and assume that $$ f\left( t \right) \sim \frac{{a_0 }} {2} + \sum\limits_{k = 1}^\infty {\left( {a_k \cos kt + b_k \sin kt} \right)} $$ Hardy and Littlewood [1] proved that the series $ \sum\limits_{k = 1}^\infty {\frac{{a_k }} {k}} $ converges if and only if the improper Riemann integral $$ \mathop {\lim }\limits_{\delta \to 0^ + } \int_\delta ^\pi {\frac{1} {x}} \left\{ {\int_{ - x}^x {f(t)dt} } \right\}dx $$ exists. In this paper we prove a refinement of this result.     

On the fast Khintchine spectrum in continued fractions

For $x\in [0,1)$ x ∈ [ 0 , 1 ) , let $x=[a_1(x), a_2(x),\ldots ]$ x = [ a 1 ( x ) , a 2 ( x ) , ... ] be its continued fraction expansion with partial quotients $\{a_n(x), n\ge 1\}$ { a n ( x ) , n ≥ 1 } . Let $\psi : \mathbb{N } \rightarrow \mathbb{N }$ ψ : N → N be a function with $\psi (n)/n\rightarrow \infty $ ψ ( n ) / n → ∞ as $n\rightarrow \infty $ n → ∞ . In this note, the fast Khintchine spectrum, i.e., the Hausdorff dimension of the set $$\begin{aligned} E(\psi ):=\left\{ x\in [0,1): \lim _{n\rightarrow \infty }\frac{1}{\psi (n)}\sum _{j=1}^n\log a_j(x)=1\right\} \end{aligned}$$ E ( ψ ) : = x ∈ [ 0 , 1 ) : lim n → ∞ 1 ψ ( n ) ∑ j = 1 n log a j ( x ) = 1 is completely determined without any extra condition on $\psi $ ψ . This fills a gap of the former work in Fan et al. (Ergod Theor Dyn Syst 29:73–109, 2009).     

Generalized monotone approximation inL p Space

Letf(x) ∈L _p[0,1], 1?p? ∞. We shall say that functionf(x)∈Δ^k (integerk?1) if for anyh ∈ [0, 1/k] andx ∈ [0,1?kh], we have Δ _h ^k f(x)?0. Denote by ∏ _n the space of algebraic polynomials of degree not exceedingn and define $$E_{n,k} (f)_p : = \mathop {\inf }\limits_{\mathop {P_n \in \prod _n }\limits_{P_n^{(\lambda )} \geqslant 0} } \parallel f(x) - P_n (x)\parallel _{L_p [0,1]} .$$ We prove that for any positive integerk, iff(x) ∈ Δ ^k ∩ L _p[0, 1], 1?p?∞, then we have $$E_{n,k} (f)_p \leqslant C\omega _2 \left( {f,\frac{1}{n}} \right)_p ,$$ whereC is a constant only depending onk.     

Stability of the Graph of a Convex Function

Let x: M → A ⁿ⁺¹ be the graph of some strongly convex function x ⁿ+1= ?( x₁,…,x_n) defined on a domain Ω ? A ⁿ in a real affine space. We consider the relative metric G, defined by $ G=\sum{\partial^{2}f\over\partial x_{i}\partial x_{j}}dx_{i}dx_{j}$ .In this paper, we calculate the second variation of the area integral with respect to the relative metric G. We prove that the parabolic affine hyperspheres are stable.     

Two results on arithmetical functions

Generalizing two results of Rieger [8] and Selberg [10] we give asymptotic formulas for sums of type $${\matrix {\sum \limits_{n\leq x}\cr n\equiv l({\rm mod}k)\cr f_{\kappa}(n)\equiv s_{\kappa}({\rm mod}p_{\kappa})\cr (\kappa=1,\dots,r)\cr}}\qquad \chi(n)\qquad {\rm and} {\matrix {\sum \limits_{n\leq x}\cr n\equiv l({\rm mod}k)\cr f_{\kappa}(n)\equiv s_{\kappa}({\rm mod}p_{\kappa})\cr (\kappa=1,\dots,r)\cr}}\qquad \chi(n),$$ where χ is a suitable multiplicative function, f₁,…, f _r are “small” additive, prime-independent arithmetical functions and k, l are coprime. The proofs are based on an analytic method which consists of considering the Dirichlet series generated by $ \chi(n)z_{1}^{f_{1}(n)}\cdot... \cdot z_{r}^{f_{r}(n)},z_{1}\dots z_{r} $ complex.     

Commutators of BMO functions and degenerate Schr?dinger operators with certain nonnegative potentials

Let ${\mathcal{L}f(x)=-\frac{1}{\omega}\sum_{i,j} \partial_i(a_{i,j}(\cdot)\partial_jf)(x)+V(x)f(x)}$ with the non-negative potential V belonging to reverse H?lder class with respect to the measure ??(x)dx, where ??(x) satisfies the A ₂ condition of Muckenhoupt and a _i,j (x) is a real symmetric matrix satisfying ${\lambda^{-1}\omega(x)|\xi|^2\le \sum^n_{i,j=1}a_{i,j}(x)\xi_i\xi_j\le\lambda\omega(x)|\xi|^2. }$ We obtain some estimates for ${V^{\alpha}\mathcal{L}^{-\alpha}}$ on the weighted L ^p spaces and we study the weighted L ^p boundedness of the commutator ${[b, V^{\alpha} \mathcal{L}^{-\alpha}]}$ when ${b\in BMO_\omega}$ and 0?<??? ?? 1.     

An Integral Jensen Inequality For Convex Multifunctions

We prove the following multivalued version of the Jensen integral inequality. Let X, Y be Banach spaces and D ? X an open and convex set. If F: D ? cl(Y) is a continuous convex function, then for each normalized measure space (Ω, S, μ), and for all μ-integrable functions ? : Ω ? D such that conv?(Ω) ? D, $$\int_{\Omega}(F\ o\ \phi)d\mu \subset F\Bigg(\int_{\Omega}\phi d\mu\Bigg).$$      

Nontrivial solutions of a higher-order rational difference equation

We prove that, for every k ∈ ?, the following generalization of the Putnam difference equation $$ x_{n + 1} = \frac{{x_n + x_{n - 1} + \cdots + x_{n - (k - 1)} + x_{n - k} x_{n - (k + 1)} }} {{x_n x_{n - 1} + x_{n - 2} + \cdots + x_{n - (k + 1)} }}, n \in \mathbb{N}_0 , $$ has a positive solution with the following asymptotics $$ x_n = 1 + (k + 1)e^{ - \lambda ^n } + (k + 1)e^{ - c\lambda ^n } + o(e^{ - c\lambda ^n } ) $$ for some c > 1 depending on k, and where λ is the root of the polynomial P(λ) = λ ^k+2 ? λ ? 1 belonging to the interval (1, 2). Using this result, we prove that the equation has a positive solution which is not eventually equal to 1. Also, for the case k = 1, we find all positive eventually equal to unity solutions to the equation.