Strict inequalities for the derivatives of functions satisfying certain boundary conditions

For functions satisfying the boundary conditions

Precise estimate of total deficiency of meromorphic derivatives

Let f(z) be a transcendental meromorphic function in the finite plane andk be a positive integer. Then we have . Moreover, if the order of f(z) is finite, then we also have , where δ(a, f^(k)) denotes the deficiency of the valuea with respect to f^(k) and θ(∞,f) is the ramification index of ∞ with respect tof.     

On inequalities for norms of intermediate derivatives on a finite interval

For functionsf which have an absolute continuous (n–1)th derivative on the interval [0, 1], it is proved that, in the case ofn>4, the inequality

Weighted a priori estimates for solution of (−Δ)<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript><Emphasis Type="Italic">u</Emphasis> = <Emphasis Type="Italic">f</Emphasis> with homogeneous dirichlet conditions

Let u be a weak solution of (-△)mu = f with Dirichlet boundary conditions in a smooth bounded domain Ω  Rn. Then, the main goal of this paper is to prove the following a priori estimate:‖u‖ Wω2 m,p(Ω) ≤ C ‖f‖ Lωp (Ω),where ω is a weight in the Muckenhoupt class Ap.     

Best polynomial approximation in Sobolev-Laguerre and Sobolev-Legendre spaces

We investigate limiting behavior as γ tends to ∞ of the best polynomial approximations in the Sobolev-Laguerre space W^N,2([0, ∞); e^−x) and the Sobolev-Legendre space W^N,2([−1, 1]) with respect to the Sobolev-Laguerre inner product

Sharp estimates for derivatives of functions in the Sobolev classes $$ \mathop {W_2^r }\limits^ \circ $$(−1,1)

Explicit formulas are obtained for the maximum possible values of the derivatives f ^(k)(x), x ∈ (−1, 1), k ∈ {0, 1, ..., r − 1}, for functions f that vanish together with their (absolutely continuous) derivatives of order up to ≤ r − 1 at the points ±1 and are such that $ \left\| {f^{\left( r \right)} } \right\|_{L_2 ( - 1,1)} \leqslant 1 $ \left\| {f^{\left( r \right)} } \right\|_{L_2 ( - 1,1)} \leqslant 1 . As a corollary, it is shown that the first eigenvalue λ _1,r of the operator (−D ²) ^r with these boundary conditions is $ \sqrt 2 $ \sqrt 2 (2r)! (1 + O(1/r)), r → ∞.     

Precise inequalities for norms of functions,third partial,second mixed,or directional derivatives

For functions f which are bounded throughout the plane R₂ together with the partial derivatives f(^3,0) f(_0,3), inequalities $$\left\| {f^{(1,1)} } \right\| \leqslant \sqrt[3]{3}\left\| f \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left\| {f^{(3,0)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left\| {f^{(0,3)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} ,\left\| {f_e^{(2)} } \right\| \leqslant \sqrt[3]{3}\left\| f \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left( {\left\| {f^{(3,0)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left| {e_1 } \right| + \left\| {f^{(0,3)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left| {e_2 } \right|} \right)^2 ,$$ are established, where ∥?∥denotes the upper bound on R² of the absolute values of the corresponding function, andf _fe ⁽²⁾ is the second derivative in the direction of the unit vector e=(e₁, e₂). Functions are exhibited for which these inequalities become equalities.     

Blow-up criterion for 2-D Boussinesq equations in bounded domain

We extend the results for 2-D Boussinesq equations from ℝ² to a bounded domain Ω. First, as for the existence of weak solutions, we transform Boussinesq equations to a nonlinear evolution equation U _t + A(t, U) = 0. In stead of using the methods of fundamental solutions in the case of entire ℝ², we study the qualities of F(u, υ) = (u · ▽)υ to get some useful estimates for A(t, U), which helps us to conclude the local-in-time existence and uniqueness of solutions. Second, as for blow-up criterions, we use energy methods, Sobolev inequalities and Gronwall inequality to control and by and . Furthermore, can control by using vorticity transportation equations. At last, can control . Thus, we can find a blow-up criterion in the form of .      

A note on curvature of α-connections of a statistical manifold

The family of α-connections ∇^(α) on a statistical manifold equipped with a pair of conjugate connections and is given as . Here, we develop an expression of curvature R ^(α) for ∇^(α) in relation to those for . Immediately evident from it is that ∇^(α) is equiaffine for any when are dually flat, as previously observed in Takeuchi and Amari (IEEE Transactions on Information Theory 51:1011–1023, 2005). Other related formulae are also developed. The work was conducted when the author was on sabbatical leave as a visiting research scientist at the Mathematical Neuroscience Unit, RIKEN Brain Science Institute, Wako-shi, Saitama 351-0198, Japan.     

Trigonometric series of classesL^p (\mathbb{T}), p \in ]1;\infty [ and their conservative meansand their conservative means

Suppose that a lower triangular matrix μ:[μ _m ⁽ⁿ⁾ ] defines a conservative summation method for series, i.e.,

A sharp form of an embedding into multiple exponential spaces

Let Θ be a bounded open set in ℝ ⁿ , n ⩾ 2. In a well-known paper Indiana Univ. Math. J., 20, 1077–1092 (1971) Moser found the smallest value of K such that

Some Inequalities for Tree Martingales

In this paper we study tree martingales and proved that if 1≤α,β〈∞,1≤p〈∞ then for every predictable tree martingale f=（ft,t∞T）and E[σ^（P）（f）]〈∞,E[S^（P）（f）]〈∞,it holds that ‖（St^（p）（f）,t∈T）‖M^α∞≤Cαβ‖f‖p^αβ,‖（σt^（p）（f）,t∈T）‖M^α,β‖f‖P^αβ,where Cαβ depends only on α and β.     

Duals of holomorphic Besov spaces on the polydisks and diagonal mappings

Let U ⁿ be the unit polydisk in C ⁿ and S be the space of functions of regular variation. Let 1 ≤ p < ∞, ω = (ω ₁, ..., ω _n ), ω _j ∈ S(1 ≤ j ≤ n) and f ∈ H(U ⁿ ). The function f is said to be in holomorphic Besov space B _p (ω) if

Exact constants in inequalities of the jackson type for quadrature formulas

We prove that if is the error of a simple quadrature formula and ω(ε, δ)₁ is the integral modulus of continuity, then, for any δ ≥/π andn,r = 1, 2, …, the following equality is true: whereD _r is the Bernoulli kernel.     

Remainder estimates for the approximation numbers of weighted Hardy operators acting onL 2

We consider the weighted Hardy integral operatorT:L ²(a, b) →L ²(a, b), −∞≤a<b≤∞, defined by . In [EEH1] and [EEH2], under certain conditions onu andv, upper and lower estimates and asymptotic results were obtained for the approximation numbersa _n(T) ofT. In this paper, we show that under suitable conditions onu andv, where ∥w∥_p=(∫ _a ^b |w(t)|^p dt)^1/p. Research supported by NSERC, grant A4021. Research supported by grant No. 201/98/P017 of the Grant Agency of the Czech Republic.     

On Sikkema-Kantorovič polynomials of order K

Let a_n≥0 and F(u)∈C [0,1], Sikkema constructed polynomials: , ifα _n ≡0, then B_n (0, F, x) are Bernstein polynomials. Let , we constructe new polynomials in this paper: Q _n ^(k) (α _n ,f(t))=d ^k /dx ^k B _n+k (α _n ,F _k (u),x), which are called Sikkema-Kantorovic polynomials of order k. Ifα _n ≡0, k=1, then Q_n ⁽¹⁾ (0, f(t), x) are Kantorovič polynomials P_n(f). Ifα _n =0, k=2, then Q_n ⁽²⁾, (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is: Theorem 2. Let 1≤p≤∞, in order that for every f∈L^P [0, 1], , it is sufficient and necessary that , § 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define^{[1–2],[8–10]}: . As usual, for the space L^p [a,b](1≤p<∞), we have and L[a, b]=l¹[a, b]. Letα _n ⩾0and F(u)∈C[0,1],Sikkema-Bernstein polynomials ^{[3] [4]}. The author expresses his thanks to Professor M. W. Müller of Dortmund University at West Germany for his supports.     

Nonparametric estimation of an affinity measure between two absolutely continuous distributions with hypotheses testing applications

LetF andG denote two distribution functions defined on the same probability space and are absolutely continuous with respect to the Lebesgue measure with probability density functionsf andg, respectively. A measure of the closeness betweenF andG is defined by: . Based on two independent samples it is proposed to estimate λ by , whereF _n (x) andG _n (x) are the empirical distribution functions ofF(x) andG(x) respectively and and are taken to be the so-called kernel estimates off(x) andg(x) respectively, as defined by Parzen [16]. Large sample theory of is presented and a two sample goodness-of-fit test is presented based on . Also discussed are estimates of certain modifications of λ which allow us to propose some test statistics for the one sample case, i.e., wheng(x)=f ₀ (x), withf ₀ (x) completely known and for testing symmetry, i.e., testingH ₀:f(x)=f(−x).     

Boundedness of parabolic singular integrals and Marcinkiewicz integrals on Triebel-Lizorkin spaces

In this paper,we obtain the boundedness of the parabolic singular integral operator T with kernel in L(logL)1/γ(Sn-1) on Triebel-Lizorkin spaces.Moreover,we prove the boundedness of a class of Marcinkiewicz integrals μΩ,q(f) from ∥f∥ F˙p0,q(Rn) into Lp(Rn).     

Strong approximation of Riesz means at critical index on Hp(T) (0<p≤1)

This paper is a continuation of [3]. Suppose f∈H^p(T), 0σ _r ^σ f,σ=1/p?1. When p=1, it is just the partial Fourier sums S_kf. In this paper we establish the sharp estimations on the degree of approximation: $$\left\{ { - \frac{1}{{logR}}\int\limits_1^R {\left\| {\sigma _r^\delta f - f} \right\|_{H^p (T)}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqq C{\mathbf{ }}{}_p\omega \left( {f,{\mathbf{ }}( - \frac{1}{{logR}})^{1/p} } \right)_{H^p (T)} ,0< p< 1,$$ and \(\frac{1}{{\log L}}\sum\limits_{k - 1}^L {\frac{{\left\| {S_k f - f} \right\|_H 1_{(T)} }}{k} \leqq Cp\omega (f; - \frac{1}{{\log L}})_H 1_{(T)} } \) Where $$\omega (f,{\mathbf{ }}h)_{H^p (T)} \begin{array}{*{20}c} { = Sup} \\ {0 \leqq \left| u \right| \leqq h} \\ \end{array} \left\| {f( \cdot + u) - f( \cdot )} \right\|_{H^p (T).} $$ .