Estimates for eigenvalues of Laplacian operator with any order

Let D be a bounded domain in an n-dimensional Euclidean space Rn. Assume that 0 ＜ λ1 ≤λ2 ≤ … ≤ λκ ≤ … are the eigenvalues of the Dirichlet Laplacian operator with any order l{(-△)lu=λu, in D u=(δ)u/(δ)(→n)=…(δ)l-1u/(δ)(→n)l-1=0,on (δ)D.Then we obtain an upper bound of the (k 1)-th eigenvalue λκ 1 in terms of the first k eigenvalues.k∑i=1(λκ 1-λi) ≤ 1/n[4l(n 2l-2)]1/2{k∑i=1(λκ 1-λi)1/2λil-1/l k∑i=1(λκ 1-λi)1/2λ1/li}1/2.This ineguality is independent of the domain D. Furthermore, for any l ≥ 3 the above inequality is better than all the known results. Our rusults are the natural generalization of inequalities corresponding to the case l = 2 considered by Qing-Ming Cheng and Hong-Cang Yang. When l = 1, our inequalities imply a weaker form of Yang inequalities. We aslo reprove an implication claimed by Cheng and Yang.     

Sharp Kolmogorov-Type Inequalities for Moduli of Continuity and Best Approximations by Trigonometric Polynomials and Splines

In what follows, $C$ is the space of -periodic continuous functions; P is a seminorm defined on C, shift-invariant, and majorized by the uniform norm; is the mth modulus of continuity of a function f with step h and calculated with respect to P; , ( ), ,

On an exponential sum

Let p be a prime number, n be a positive integer, and ƒ(x) = ax^k + bx. We put

Approximating periodic functions by interpolation sums of Jackson type

Let

The true order of magnitude of Lamé rounding error sums

Let (z ∈ ℝ). Further let λ denote a large real parameter. We show that for arbitrary real numbersk and α withk>=2.7013 and 0<α≦1,

多线性分数次积分算子交换子的有界性

Ibαf ( x) =∫R ∏mj=1( bj( x) - bj( y) ) 1| x - y| n-αf ( y) dyare considered.The following priori estimates are proved.For 1 01Φ1t| {y∈Rn:| Ibαf( y) | >t}| 1q ≤csupt>01Φ1t| {y∈Rn:ML( log L) 1r ,α(‖b‖f ) ( y) >t}| 1q,where‖b‖=∏mj=1‖bj‖Oscexp Lrj,Φ( t) =t( 1 + log+t) 1r,1r =1r1+ ...+ 1rm,ML(…     

Nikolskii type inequality for cardinal splines

The Nikolskii type inequality for cardinal splines

An ultimate extremely accurate formula for approximation of the factorial function

We prove in this paper that for every x ≥ 0,

An end point estimate for maximal commutators

In this paper we prove that the maximal commutator of singular integral operator [b, T]^* satisfies the inequality:

Sums of Five, Seven and Nine Squares

Let r _k(n) denote the number of representations of an integer n as a sum of k squares. We prove that

An Elementary Proof of Pólya-Vinogradov's Inequality, II

Let be a primitive character mod k, k > 2. In [1], the following elementary estimate

On the Relationship Between the Baum-Katz-Spitzer Complete Convergence Theorem and the Law of the Iterated Logarithm

For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of the iterated logarithm is investigated. Sets of conditions are provided under which （i） lim sup n→∞ ｜｜Sn｜｜/an〈∞ a.s.and ∞ ∑n=1（1/n）P（｜｜Sn｜｜/an ≥ε〈∞for all ε 〉 λ for some constant λ ∈ [0, ∞） are equivalent; （ii） For all constants λ ∈ [0, ∞）, lim sup ｜｜Sn｜｜/an =λ a.s.and ^∞∑ n=1（1/n） P（｜｜Sn｜｜/an ≥ε）{〈∞, if ε〉λ =∞,if ε〈λare equivalent. In general, no geometric conditions are imposed on the underlying Banach space. Corollaries are presented and new results are obtained even in the case of real-valued random variables.     

On the riemann zeta-function and the divisor problem II

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If E ^*(t)=E(t)-2πΔ^*(t/2π) with , then we obtain

一类时滞Logistic差分方程的全局吸引性

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｎｏｎａｕｔｏｎｏｍｏｕｓｄｅｌａｙｌｏｇｉｓｔｉｃｄｉｆｆｅｒｅｎｃｅｅｑｕａｔｉｏｎΔｙｎ =ｐｎｙｎ( 1 - ｙτ(ｎ) )　 ,ｎ =0 ,1 ,2 ,...,( 1 1 )ｗｈｅｒｅｐｎ ∞ｎ =0 ｉｓａｓｅｑｕｅｎｃｅｏｆｐｏｓｉｔｉｖｅｒｅａｌｎｕｍｂｅｒｓ ,τ(ｎ) ∞ｎ =0 ｉｓａｎｏｎｄｅｃｒｅａｓｉｎｇｓｅｑｕｅｎｃｅｏｆｉｎｔｅｇｅｒｓ,τ(ｎ) <ｎａｎｄｌｉｍｎ→∞τ(ｎ) =∞ ,Δｙｎ=ｙｎ +1- ｙｎ.Ｍｏｔｉｖａｔｅｄｂｙｐｌａｕｓｉｂｌｅａｐｐｌｉｃａｔｉｏｎｓ…     

Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity

We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space

Bounds for the Third Peano Constants of Gaussian Quadrature Formulae

Denote by R _n ^G the remainder functional of the Gaussian quadrature formula involving n nodes. Using an elementary method, we derive the asymptotically best possible inequalities

Cubic elliptic functions

The function

Two classes of finite semigroups and monoids involving Lucas numbers

The class of finitely presented groups is an extension of the class of triangle groups studied recently. These groups are finite and their orders depend on the Lucas numbers. In this paper, by considering the three presentations

Approximating periodic functions in Hölder type metrics by singular integrals with positive Kernels

Let M be either the space of 2π-periodic functions L_p, where 1 ≤ p < ∞, or C; let ω_r(f, h) be the continuity modulus of order r of the function f, and let