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1.
Given a nontrivial Borel measure on ℝ, let p n be the corresponding orthonormal polynomial of degree n whose zeros are λ j (n), j=1,…,n. Then for each j=1,…,n,
with
defines a discrete probability distribution. The Shannon entropy of the sequence {p n } is consequently defined as
In the case of Chebyshev polynomials of the first and second kinds, an explicit and closed formula for is obtained, revealing interesting connections with number theory. In addition, several results of numerical computations exemplifying the behavior of for other families are presented.   相似文献   

2.
Let A 0, ... , A n−1 be operators on a separable complex Hilbert space , and let α0,..., α n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm,
for 2 ≤ p < ∞, and the reverse inequality holds for 0 < p ≤ 2. Moreover, we prove that if ω0,..., ω n−1 are the n roots of unity with ω j = e 2πij/n , 0 ≤ j ≤ n − 1, then for every unitarily invariant norm,
for 2 ≤ p < ∞, and the reverse inequalities hold for 0 < p ≤ 2. These inequalities, which involve n-tuples of operators, lead to natural generalizations and refinements of some of the classical Clarkson inequalities in the Schatten p-norms. Extensions of these inequalities to certain convex and concave functions, including the power functions, are olso optained.   相似文献   

3.
We consider a class of fourth-order nonlinear difference equations of the form
where α and β are the ratios of odd positive integers, and {p n } and {q n } are positive real sequences defined for all satisfying the condition
We classify the nonoscillatory solutions of (Ω) and establish necessary and/or sufficient conditions for the existence of nonoscillatory solutions with specific asymptotic behavior. Supported by Ministry of Science, Technology and Development of Republic of Serbia – Grant No. 144003.  相似文献   

4.
Suppose that 0<δ≤1,N=1/δ, and α, ga≥0, is an integer. For the classical Meixner polynomials orthonormal on the gird {0, δ, 2δ, ...} with weight ρ(x)=(1-e −δ)αг(Nx+α+ 1)/г(Nx+1), the following asymptotic formula is obtained: . The remainderv n,N α (z) forn≤λN satisfies the estimate
where Λ k α (x) are the Laguerre orthonormal polynomials. As a consequence, a weighted estimate, for the Meixner polynomial on the semiaxis [0, ∞) is obtained. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 603–616, October, 1997. Translated by N. K. Kulman  相似文献   

5.
We investigate the boundary growth of positive superharmonic functions u on a bounded domain Ω in , n ≥ 3, satisfying the nonlinear elliptic inequality
where c >  0, α ≥ 0 and p >  0 are constants, and is the distance from x to the boundary of Ω. The result is applied to show a Harnack inequality for such superharmonic functions. Also, we study the existence of positive solutions, with singularity on the boundary, of the nonlinear elliptic equation
where V and f are Borel measurable functions conditioned by the generalized Kato class.  相似文献   

6.
We consider some classes of 2π-periodic functions defined by a class of operators having certain oscillation properties, which include the classical Sobolev class and a class of analytic functions which can not be represented as a convolution class as its special cases. Let be the largest integer not bigger than x. We prove that on these classes of functions the rectangular formula
is optimal among all quadrature formulae of the form
where the nodes 0 ≤  t 1 < ... < t n  < 2π and the coefficients (weights) are arbitrary, i = 1,...,n, j = 0,1,..., ν i − 1, and (ν1,...,ν n ) is a system of positive integers satisfying the condition . In particular, the rectangular formula is optimal for these classes of functions among all quadrature formulae of the form
with free nodes 0 ≤  t 1 <  ... < t N <  2π and arbitrary weights . Moreover, we exactly determine the error estimates of the optimal quadrature formulae on these classes of functions.Project supported by the National Natural Science Foundation of China (Grant No. 10671019) and Research Fund for the Doctoral Program Higher Education (Grant No. 20050027007).  相似文献   

7.
Suppose thatG is a finitely connected domain with rectifiable boundary γ, ∞εG, the domainsD 1,...,D s are the complements ofG, the subsetsF j ⊂D j are infinite and compact,n j ≥1,j=1,...,s, are integers, λ0 is a complex-valued measure on γ, and
We consider the extremum problem
where μ j ,j=1,...,s, are complex-valued measures onF j and
are Golubev sums. We prove that β=Δ, where
We also establish several other relations between these and other extremal variables. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 738–745, May, 1999.  相似文献   

8.
Let n≥2 be an integer number. In this paper, we investigate the generalized Hyers Ulam- Rassias stability in Banach spaces and also Banach modules over a Banach algebra and a C＊-algebra and the stability using the alternative fixed point of an n-dimensional cubic functional equation in Banach spaces：f（2∑j=1^n-1 xj＋xn）＋f（2∑j=1^n-1 xj-xn）＋4∑j=1^n-1f（xj）=16f（∑j=1^n-1 xj）＋2∑j=1^n-1（f（xj＋xn）＋f（xj-xn）  相似文献   

9.
For a family of compact Riemann surfaces Xt of genus g > 1, parameterized by the Schottky space we define a natural basis of which varies holomorphically with t and generalizes the basis of normalized abelian differentials of the first kind for n  =  1. We introduce a holomorphic function F(n) on which generalizes the classical product for n  =  1 and g  =  1. We prove the holomorphic factorization formula
where det'Δ n is the zeta-function regularized determinant of the Laplace operator Δn in the hyperbolic metric acting on n-differentials, Nn is the Gram matrix of the natural basis with respect to the inner product given by the hyperbolic metric, S is the classical Liouville action –a K?hler potential of the Weil–Petersson metric on – and cg,n is a constant depending only on g and n. The factorization formula reduces to Kronecker’s first limit formula when n  =  1 and g  =  1, and to Zograf’s factorization formula for n  =  1 and g  >  1. Received: April 2005. Accepted: October 2005  相似文献   

10.
§ 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…  相似文献   

11.
Let n–1 be the linear space of algebraic polynomials of degreen–1. We prove that the extremal problem
  相似文献   

12.
All the letters represent relative integers, except and and i = e1e2 in R(2, 0) oder e1 in R(1, 0). We study the Fermat’s equation
(1)
a, b, c being prime two and two and by utilizing an elementary method. We use the Gauss’ formula
where n = 5, 7, 11, 17.
1.  If 2 is the p · g · c · d· of A and B we put
and A′ and B′ are prime between themselves.
2.  If βn = bn / (c − a) is not divisible by n, we write the expansion
(2)
by puting oder whereas oder It follows that B ′ is divisible by n.
3.  If one a, b oder c is divisible by n we prove the impossibility
4.  In the case n = 3 the ring {a + bj} is euclidian which permits to conclude in favour of the impossibility.
  相似文献   

13.
We consider the following Liouville equation in
For each fixed and a j  > 0 for 1 ≤ j ≤ k, we construct a solution to the above equation with the following asymptotic behavior:
  相似文献   

14.
Nonimprovable effective sufficient conditions are established for the unique solvability of the periodic problem
where ω  >  0, ℓi : C([0, ω])→ L([0,ω]) are linear bounded operators, and qi∈L([0, ω]). Received: 11 June 2005  相似文献   

15.
Let τ(n) be the Ramanujan τ-function, x ≥ 10 be an integer parameter. We prove that
We also show that
where ω(n) is the number of distinct prime divisors of n and p denotes prime numbers. These estimates improve several results from [6, 9]. Received: 23 November 2006  相似文献   

16.
For an integer m ≥ 4, we define a set of 2[m/2] × 2[m/2] matrices γj （m）, （j = 0, 1,..., m - 1） which satisfy γj （m）γk （m） ＋γk （m）γj （m） = 2ηjk （m）I[m/2], where （ηjk （m）） 0≤j,k≤m-1 is a diagonal matrix, the first diagonal element of which is 1 and the others are -1, I[m/2] is a 2[m/1] × 2[m/2] identity matrix with [m/2] being the integer part of m/2. For m = 4 and 5, the representation （m） of the Lorentz Spin group is known. For m≥ 6, we prove that （i） when m = 2n, （n ≥ 3）, （m） is the group generated by the set of matrices {T｜T=1/√ξ（（I＋k） 0 ＋ 0 I-K） （ U 0 0 U）, （ii） when m = 2n ＋ 1 （n≥ 3）, （m） is generated by the set of matrices {T｜T=1/√ξ（I -k^- k I）U,U∈ （m-1）,ξ=1-m-2 ∑k,j=0 ηkja^k a^j〉0, K=i[m-3 ∑j=0 a^j γj（m-2）＋a^（m-2） In],K^-=i[m-3∑j=0 a^j γj（m-2）-a^（m-2） In]}  相似文献   

17.
Our first basic model is the fully nonlinear dual porous medium equation with source
for which we consider the Cauchy problem with given nonnegative bounded initial data u0. For the semilinear case m=1, the critical exponent was obtained by H. Fujita in 1966. For p ∈(1, p0] any nontrivial solution blows up in finite time, while for p > p0 there exist sufficiently small global solutions. During last thirty years such critical exponents were detected for many semilinear and quasilinear parabolic, hyperbolic and elliptic PDEs and inequalities. Most of efforts were devoted to equations with differential operators in divergent form, where classical techniques associated with weak solutions and integration by parts with a variety of test functions can be applied. Using this fully nonlinear equation, we propose and develop new approaches to calculating critical Fujita exponents in different functional settings. The second models with a “semi-divergent” diffusion operator is the thin film equation with source
for which the critical exponent is shown to be   相似文献   

18.
Multilinear Singular Integrals with Rough Kernel   总被引：9，自引：0，他引：9  
For a class of multilinear singular integral operators T A ,
where R m (A; x, y) denotes the m-th Taylor series remainder of A at x expanded about y, A has derivatives of order m − 1 in is homogeneous of degree zero, the authors prove that T A is bounded from L p (ℝ n ) to and from L 1(ℝ n ) to L n/(n−β),∞(ℝ n ) with the bound And if Ω has vanishing moments of order m − 1 and satisfies some kinds of Dini regularity otherwise, then T A is also bounded from L p (ℝ n ) to with the bound Supported by the National 973 Project (G1990751) and SEDF of China (20010027002)  相似文献   

19.
Given k ∈ L1 (0,1) satisfying certain smoothness and growth conditions at 0, we consider the Volterra convolution operator Vk defined on Lp (0,1) by
and its iterates We construct some much simpler sequences which, as n → ∞, are asymptotically equal in the operator norm to Vkn. This leads to a simple asymptotic formula for ||Vkn|| and to a simple ‘asymptotically extremal sequence’; that is, a sequence (un) in Lp (0, 1) with ||un||p=1 and as n → ∞. As an application, we derive a limit theorem for large deviations, which appears to be beyond the established theory.  相似文献   

20.
Let Sn = X1 + · · · + X n , n ≥ 1, and S 0 = 0, where X 1, X 2, . . . are independent, identically distributed random variables such that the distribution of S n/B n converges weakly to a nondeoenerate distribution F α as n → ∞ for some positive B n . We study asymptotic behavior of sums of the form
where
a function d(t) is continuous at [0,1] and has power decay at zero,
Bibliography: 13 titles. Translated from Zapiski Nauchnylch Serninarov POMI, Vol. 361, 2008, pp. 109–122.  相似文献   

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