共查询到20条相似文献,搜索用时 46 毫秒
1.
A. I. Aptekarev J. S. Dehesa A. Martínez-Finkelshtein R. Yáñez 《Constructive Approximation》2009,30(1):93-119
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.
J. V. Manojlović 《Lithuanian Mathematical Journal》2009,49(1):71-92
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.
I. I. Sharapudinov 《Mathematical Notes》1997,62(4):501-512
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.
The boundary growth of superharmonic functions and positive solutions of nonlinear elliptic equations 总被引:1,自引:0,他引:1
Kentaro Hirata 《Mathematische Annalen》2008,340(3):625-645
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.
S. Ya. Khavinson 《Mathematical Notes》1999,65(5):620-626
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.
Holomorphic factorization of determinants of Laplacians on Riemann surfaces and a higher genus generalization of Kronecker’s first limit formula 总被引:2,自引:0,他引:2
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.
周英告 《高校应用数学学报(英文版)》2003,18(1):53-58
§ 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.
D. P. Dryanov 《Constructive Approximation》1994,10(3):377-409
Let
n–1 be the linear space of algebraic polynomials of degreen–1. We prove that the extremal problem
相似文献
12.
Gaston Casanova 《Advances in Applied Clifford Algebras》2005,15(1):151-155
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
13.
We consider the following Liouville equation in
14.
On a periodic boundary value problem for cyclic feedback type linear functional differential systems
Sulkhan Mukhigulashvili 《Archiv der Mathematik》2006,87(3):255-260
Nonimprovable effective sufficient conditions are established for the unique solvability of the periodic problem
15.
Let τ(n) be the Ramanujan τ-function, x ≥ 10 be an integer parameter. We prove that
16.
Qi Keng LU Ke WU 《数学学报(英文版)》2007,23(4):577-598
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
18.
Multilinear Singular Integrals with Rough Kernel 总被引:9,自引:0,他引:9
ShanZhenLU HuoXiongWU PuZHANG 《数学学报(英文版)》2003,19(1):51-62
For a class of multilinear singular integral operators T
A
,
19.
S. P. Eveson 《Integral Equations and Operator Theory》2005,53(3):331-341
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
20.
L. V. Rozovsky 《Journal of Mathematical Sciences》2009,159(3):341-349
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
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