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 $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\frac{{\sqrt {1 - x^2 } }} {{n^r }}\left| {T_n (x)} \right|\omega \left( {f^{(r)} ,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left( {\left( {\sqrt {1 - x^2 } + \frac{1} {n}} \right)^{r - 1} \ln (n + 1) + 1} \right)} \right\}. $$      

A note on strong summability of two-dimensional Walsh-Fourier series

The paper deals with the strong summability of Marcinkiewicz means with a variable power. Let $$H_n \left( {f,x,y,A_n } \right): = \tfrac{1} {n}\sum\nolimits_{l = 1}^n {\left( {e^{\left. {A_n } \right|\left. {S_{ll} \left( {f,x,y} \right) - f\left( {x,y} \right)} \right|^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } - 1} \right)} .$$ It is shown that if A _n ↑ ∞ arbitrary slowly, there exists f ∈ C(I ²) such that lim _n→∞ H _n (f, 0, 0, A _n ) = +∞. At the same time, for every f ∈ C (I ²) there exists A _n (f) ↑ ∞ such that lim _n→∞ H _n (f, x, y, A _n ) = 0 uniformly on I ².     

Simultaneous approximation by algebraic polynomials

Some estimates for simultaneous polynomial approximation of a function and its derivatives are obtained. These estimates are exact in a certain sense. In particular, the following result is derived as a corollary: Forf∈C ^r[?1,1],m∈N, and anyn≥max{m+r?1, 2r+1}, an algebraic polynomialP _n of degree ≤n exists that satisfies $$\left| {f^{\left( k \right)} \left( x \right) - P_n^{\left( k \right)} \left( {f,x} \right)} \right| \leqslant C\left( {r,m} \right)\Gamma _{nrmk} \left( x \right)^{r - k} \omega ^m \left( {f^{\left( r \right)} ,\Gamma _{nrmk} \left( x \right)} \right),$$ for 0≤k≤r andx ∈ [?1,1], where ω^υ(f^(k),δ) denotes the usual vth modulus of smoothness off ^(k), and Moreover, for no 0≤k≤r can (1?x ²)( ^{r?k+1)/(r?k+m)}(1/n²)^{(m?1)/(r?k+m)} be replaced by (1-x²)^αkn^2αk-2, with α_k>(r-k+a)/(r-k+m).     

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.     

Estimation of the remainder of a cubature formula on a Chebyshev grid

Let C(Q) denote the space of continuous functions f(x, y) in the square Q = [?1, 1] × [?1, 1] with the norm $\begin{gathered} \left\| f \right\| = \max \left| {f(x,y)} \right|, \hfill \\ (x,y) \in Q. \hfill \\ \end{gathered} $ On a Chebyshev grid, a cubature formula of the form $\int\limits_{ - 1}^1 {\int\limits_{ - 1}^1 {\frac{1} {{\sqrt {(1 - x^2 )(1 - y^2 )} }}f(x,y)dxdy = \frac{{\pi ^2 }} {{mn}}\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^m {f\left( {\cos \frac{{2i - 1}} {{2n}}\pi ,\cos \frac{{2j - 1}} {{2m}}\pi } \right)} + R_{m,n} (f)} } } $ is considered in some class H(r ₁, r ₂) of functions f ?? C(Q) defined by a generalized shift operator. The remainder R _{m, n} (f) is proved to satisfy the estimate $\mathop {\sup }\limits_{f \in H(r_1 ,r_2 )} \left| {R_{m,n} (f)} \right| = O(n^{ - r_1 + 1} + m^{ - r_2 + 1} ), $ where r ₁, r ₂ > 1; ??^?1 ?? n/m ?? ?? with ?? > 0; and the constant in O(1) depends on ??.     

The “Full Müntz Theorem” inL p[0, 1] for 0<p<∞

Denote by span {f ₁,f ₂, …} the collection of all finite linear combinations of the functionsf ₁,f ₂, … over ?. The principal result of the paper is the following. Theorem (Full Müntz Theorem in L_p(A) for p ∈ (0, ∞) and for compact sets A ? [0, 1] with positive lower density at 0). Let A ? [0, 1] be a compact set with positive lower density at 0. Let p ∈ (0, ∞). Suppose (λ _j ) _j=1 ^∞ is a sequence of distinct real numbers greater than ?(1/p). Then span {x ^λ1,x ^λ2,…} is dense in L_p(A) if and only if $\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $ . Moreover, if $\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $ , then every function from the L_p(A) closure of {x ^λ1,x ^λ2,…} can be represented as an analytic function on {z ∈ ? \ (?∞,0] : |z| < r_A} restricted to A ∩ (0, r_A) where $r_A : = \sup \left\{ {y \in \mathbb{R}:\backslash ( - \infty ,0]:\left| z \right|< r_A } \right\}$ (m(·) denotes the one-dimensional Lebesgue measure). This improves and extends earlier results of Müntz, Szász, Clarkson, Erdös, P. Borwein, Erdélyi, and Operstein. Related issues about the denseness of {x ^λ1,x ^λ2,…} are also considered.     

Construction of spherical 4- and 5-designs

A finite subsetX of thed-dimensional unit sphereS ^d-1 is called a sphericalt-design, if and only if $$\frac{1}{{\left| {S^{d - 1} } \right|}}\int_{S^{d - 1} } {f(x)d\omega (x)} = \frac{1}{{\left| x \right|}}\sum\limits_{x \in X} {f(x)} $$ holds for all polynomialsf(x) =f(x ₁,x ₂,...,x _d ) of degree at mostt. In 1984 Seymour and Zaslavsky proved the existence of sphericalt-designs for anyt andd, but for sufficiently large |X|. Since spherical designs can be used for numerical integration, it is of interest to give explicit constructions. Mimura gave a construction fort = 2,d ∈ ? and |X| ≥n ₂ for somen ₂ ∈ ? (n ₂ is sharp). Here we will give an explicit construction fort = 4 and 5,d ∈ ? and |X| ≥n ₄ for somen ₄ ∈ ?.     

Commutators of Marcinkiewicz integral with rough kernels on Sobolev spaces

In this paper, the authors give the boundedness of the commutator [b, ??_??,?? ] from the homogeneous Sobolev space $\dot L_\gamma ^p \left( {\mathbb{R}^n } \right)$ to the Lebesgue space L ^p (? ⁿ ) for 1 < p < ??, where ??_??,?? denotes the Marcinkiewicz integral with rough hypersingular kernel defined by $\mu _{\Omega ,\gamma } f\left( x \right) = \left( {\int_0^\infty {\left| {\int_{\left| {x - y} \right| \leqslant t} {\frac{{\Omega \left( {x - y} \right)}} {{\left| {x - y} \right|^{n - 1} }}f\left( y \right)dy} } \right|^2 \frac{{dt}} {{t^{3 + 2\gamma } }}} } \right)^{\frac{1} {2}} ,$ , with ?? ?? L ¹(S ^n?1) for $0 < \gamma < min\left\{ {\frac{n} {2},\frac{n} {p}} \right\}$ or ?? ?? L(log⁺ L) ^?? (S ^n?1) for $\left| {1 - \frac{2} {p}} \right| < \beta < 1\left( {0 < \gamma < \frac{n} {2}} \right)$ , respectively.     

Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials

We prove that max |p′(x)|, where p runs over the set of all algebraic polynomials of degree not higher than n ≥ 3 bounded in modulus by 1 on [−1, 1], is not lower than ( n - 1 ) \mathord

Generalized n-Laplacian: Semilinear Neumann problem with the critical growth

Let Ω ? ? ⁿ , n ? 2, be a bounded connected domain of the class C ^1,θ for some θ ∈ (0, 1]. Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem $$\begin{gathered} u \in W^1 L^\Phi \left( \Omega \right), - div\left( {\Phi '\left( {\left| {\nabla u} \right|} \right)\frac{{\nabla u}} {{\left| {\nabla u} \right|}}} \right) + V\left( x \right)\Phi '\left( {\left| u \right|} \right)\frac{u} {{\left| u \right|}} = f\left( {x,u} \right) + \mu h\left( x \right) in \Omega , \hfill \\ \frac{{\partial u}} {{\partial n}} = 0 on \partial \Omega , \hfill \\ \end{gathered}$$ where Φ is a Young function such that the space W ¹ L ^Φ(Ω) is embedded into exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V (x) is a continuous potential, h ∈ (L ^Φ(Ω))* is a nontrivial continuous function, µ ? 0 is a small parameter and n denotes the outward unit normal to ?Ω.     

One Approach to the Computation of Asymptotics of Integrals of Rapidly Varying Functions

We consider integrals of the form

$$I\left( {x,h} \right) = \frac{1}{{{{\left( {2\pi h} \right)}^{k/2}}}}\int_{{\mathbb{R}^k}} {f\left( {\frac{{S\left( {x,\theta } \right)}}{h},x,\theta } \right)} d\theta $$

, where h is a small positive parameter and S(x, θ) and f(τ, x, θ) are smooth functions of variables τ ∈ ?, x ∈ ? ⁿ , and θ ∈ ? ^k ; moreover, S(x, θ) is real-valued and f(τ, x, θ) rapidly decays as |τ| →∞. We suggest an approach to the computation of the asymptotics of such integrals as h → 0 with the use of the abstract stationary phase method.

     

An initial value problem for a singular parabolic equation of even order

At first Cauchy-problem for the equation: \(L[u(X,t)] \equiv \sum\limits_{i = 1}^n {\frac{{\partial ^2 u}}{{\partial x_1^2 }} + \frac{{2v}}{{\left| X \right|^2 }}} \sum\limits_{i = 1}^n {x_i \frac{{\partial u}}{{\partial x_i }} - \frac{{\partial u}}{{\partial t}} = 0} \) wheren≥1,v—an arbitrary constant,t>0,X=(x ₁, …, x_n)∈E ⁿ/{0}, |X|= =(x ₁ ² +…+x _n ² )^1/2, with 0 being a centre of coordinate system, is studied. Basing on the above, the solution of Cauchy-Nicolescu problem is given which consist in finding a solution of the equationL ^p [u (X, t)]=0, withp∈N subject the initial conditions \(\mathop {\lim }\limits_{t \to \infty } L^k [u(X,t)] = \varphi _k (X)\) ,k=0, 1,…,p?1 and ?_k(X) are given functions.     

On the best constants in the Khintchine inequality for Steinhaus variables

Let S _j : (Ω, P) → S ¹ ? ? be an i.i.d. sequence of Steinhaus random variables, i.e. variables which are uniformly distributed on the circle S ¹. We determine the best constants a _p in the Khintchine-type inequality $${a_p}{\left\| x \right\|_2} \leqslant {\left( {{\text{E}}{{\left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|}^p}} \right)^{1/p}} \leqslant {\left\| x \right\|_2};{\text{ }}x = ({x_j})_{j = 1}^n \in {{\Bbb C}^n}$$ for 0 < p < 1, verifying a conjecture of U. Haagerup that $${a_p} = \min \left( {\Gamma {{\left( {\frac{p}{2} + 1} \right)}^{1/p}},\sqrt 2 {{\left( {{{\Gamma \left( {\frac{{p + 1}}{2}} \right)} \mathord{\left/ {\vphantom {{\Gamma \left( {\frac{{p + 1}}{2}} \right)} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right)}^{1/p}}} \right)$$ . Both expressions are equal for p = p ₀ }~ 0.4756. For p ≥ 1 the best constants a _p have been known for some time. The result implies for a norm 1 sequence x ∈ ? ⁿ , ‖x‖₂ = 1, that $${\text{E}}\ln \left| {\frac{{{S_1} + {S_2}}}{{\sqrt 2 }}} \right| \leqslant {\text{E}}\ln \left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|$$ , answering a question of A. Baernstein and R. Culverhouse.     

Refinement of an inequality of P. L. Chebyshev

Let P _n denote the linear space of polynomials p(z:=Σ _k=0 ⁿ a _k (p)z ^k of degree ≦ n with complex coefficients and let |p|_[?1,1]: = max _x∈[?1,1]|p(x)| be the uniform norm of a polynomial p over the unit interval [?1, 1]. Let t _n ∈ P _n be the n ^th Chebyshev polynomial. The inequality $$ \frac{{\left| p \right|_{\left[ { - 1,1} \right]} }} {{\left| {a_n (p)} \right|}} \geqq \frac{{\left| {t_n } \right|_{\left[ { - 1,1} \right]} }} {{\left| {a_n (t_n )} \right|}},p \in P_n $$ due to P. L. Chebyshev can be considered as an extremal property of the Chebyshev polynomial t _n in P _n . The present note contains various extensions and improvements of the above inequality obtained by using complex analysis methods.     

Estimate of a sum of Legendre symbols of polynomials of even degree

Let n≥4 be even, p > (n²?2n)/2 be simple odd, andf(x)=a ₀+a ₁+...+a _nxⁿ be a polynomial with integral coefficients that are not quadratic over the residue field modulo p, (a _n, p)=1. The following inequality is proved: $$\left| {\sum\nolimits_{x = 1}^p {\left( {\frac{{f(x)}}{p}} \right)} } \right| \leqslant (n - 2)\sqrt {p + 1 - \frac{{n(n - 4)}}{4}} + 1.$$      

Aubry–Mather Sets for Relativistic Oscillators with Anharmonic Potentials

In this paper, by the Aubry–Mather theory, it is proved that there are many periodic solutions and usual or generalized quasiperiodic solutions for relativistic oscillator with anharmonic potentials models d/dt(x/(1-|x|~2~(1/2))+ |x|~(α-1)x=p(t),where p(t) ∈ C~0(R~1) is 1-periodic and α 0.     

Estimates of maximal functions measuring local smoothness

Letη be a nondecreasing function on (0, 1] such thatη(t)/t decreases andη(+0)=0. Letf ∈L(I ⁿ ) (I≡[0,1]. Set $${\mathcal{N}}_\eta f(x) = \sup \frac{1}{{\left| Q \right|\eta (\left| Q \right|^{1/n} )}} \smallint _Q \left| {f(t) - f(x)} \right|dt,$$ , where the supremum is taken over all cubes containing the pointx. Forη=t ^α (0<α≤1) this definition was given by A.Calderón. In the paper we prove estimates of the maximal functions ${\mathcal{N}}_\eta f$ , along with some embedding theorems. In particular, we prove the following Sobolev type inequality: if $$1 \leqslant p< q< \infty , \theta \equiv n(1/p - 1/q)< 1, and \eta (t) \leqslant t^\theta \sigma (t),$$ , then $$\parallel {\mathcal{N}}_\sigma {f} {\parallel_{q,p}} \leqslant c \parallel {\mathcal{N}}_\eta {f} {\parallel_p} .$$ . Furthermore, we obtain estimates of ${\mathcal{N}}_\eta f$ in terms of theL ^p -modulus of continuity off. We find sharp conditions for ${\mathcal{N}}_\eta f$ to belong toL ^p (I ⁿ ) and the Orlicz class?(L), too.     

On the existence and smoothness of a generalized solution of a nonlinear differential equation with degeneration

Let Ω be an arbitrary open set in R _n , and let σ(x) and g _i (x), i = 1, 2, ..., n, be positive functions in Ω. We prove a embedding theorem of different metrics for the spaces W _p ^r (Ω, σ, $ \vec g $ ), where r ∈ N, p ≥ 1, and $ \vec g $ (x) = (g ₁(x), g ₂(x), ..., g _n (x)), with the norm $$ \left\| {u;W_p^r (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\left\| {u;L_{p,r}^r (\Omega ;\sigma ,\vec g)} \right\|^p + \left\| {u;L_{p,r}^0 (\Omega ;\sigma ,\vec g)} \right\|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ where $$ \left\| {u;L_{p,r}^m (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\sum\limits_{\left| k \right| = m} {\int\limits_\Omega {(\sigma (x)g_1^{k_1 - r} (x)g_2^{k_2 - r} (x) \cdots g_n^{k_n - r} (x)\left| {u^{(k)} (x)} \right|)^p dx} } } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ We use this theorem to prove the existence and uniqueness of a minimizing element U(x) ∈ W _p ^r (Ω, σ, $ \vec g $ ) for the functional $$ \Phi (u) = \sum\limits_{\left| k \right| \leqslant r} {\frac{1} {{p_k }}\int\limits_\Omega {a_k (x)} \left| {u^{(k)} (x)} \right|^{p_k } } dx - \left\langle {F,u} \right\rangle , $$ where F is a given functional. We show that the function U(x) is a generalized solution of the corresponding nonlinear differential equation. For the case in which Ω is bounded, we study the differential properties of the generalized solution depending on the smoothness of the coefficients and the right-hand side of the equation.     

Quasi-convex Mappings on the Unit Polydisk in $\mathbb{C}^{n}$

In this paper, the sharp estimates of all homogeneous expansions for f are established, where f(z) = (f ₁(z), f ₂(z), …, f _n (z))′ is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in ℂ ⁿ and

L p-approximation by Kantorovič operators

Оператор Канторович а дляf∈L _p(I), I=[0,1], определяе тся соотношением $$P_n (f,x) = (n + 1)\sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)} x^k (1 - x)^{n - 1} \int\limits_{I_k } {f(t)dt,} $$ гдеI _k=[k/(n}+1),(k+1)/(n+ 1)],n∈N. Доказывается, что есл ир>1 иf∈W _p ² (I), т.е.f абсол ютно непрерывна наI иf″∈L _p(I), то $$\left\| {P_n f - f} \right\|_p = O(n^{ - 1} ).$$ Далее, установлено, чт о еслиf∈L _p(I),p>1 и ∥P _n f-f∥_р=О(n ^?1), тоf∈S, гдеS={f∶f аб-солютно непрерывна наI, x(1?x)f′(x)=∝ ₀ ^x h(t)dt, гдеh∈L _p(I) и ∝ ₀ ¹ h(t)dt=0}. Если жеf∈L_p(I),p>1, то из условия ∥P _n(f)?f∥_pL=o(n^?1) вытекает, чтоf постоянна почти всюду.