共查询到20条相似文献,搜索用时 15 毫秒
1.
Piotr Niemiec 《Rendiconti del Circolo Matematico di Palermo》2008,57(3):391-399
The aim of the paper is to prove that every f ∈ L
1([0,1]) is of the form f = , where j
n,k
is the characteristic function of the interval [k- 1 / 2
n
, k / 2
n
) and Σ
n=0∞Σ
k=12n
|a
n,k
| is arbitrarily close to ||f|| (Theorem 2). It is also shown that if μ is any probabilistic Borel measure on [0,1], then for any ɛ > 0 there exists a sequence (b
n,k
)
n≧0
k=1,...,2n
of real numbers such that and for each Lipschitz function g: [0,1] → ℝ (Theorem 3).
相似文献
2.
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
3.
A class Uk1 (J){\mathcal{U}}_{\kappa 1} (J) of generalized J-inner mvf’s (matrix valued functions) W(λ) which appear as resolvent matrices for bitangential interpolation problems in the generalized Schur class of p ×q mvf¢s Skp ×qp \times q \, {\rm mvf's}\, {\mathcal{S}}_{\kappa}^{p \times q} and some associated reproducing kernel Pontryagin spaces are studied. These spaces are used to describe the range of the
linear fractional transformation TW based on W and applied to Sk2p ×q{\mathcal{S}}_{\kappa 2}^{p \times q}. Factorization formulas for mvf’s W in a subclass U°k1 (J) of Uk1(J){\mathcal{U}^{\circ}_{\kappa 1}} (J)\, {\rm of}\, {\mathcal{U}}_{\kappa 1}(J) found and then used to parametrize the set Sk1+k2p ×q ?TW [ Sk2p ×q ]{\mathcal{S}}_{{\kappa 1}+{\kappa 2}}^{p \times q} \cap T_{W} \left[ {\mathcal{S}}_{\kappa 2}^{p \times q} \right]. Applications to bitangential interpolation problems in the class Sk1+k2p ×q{\mathcal{S}}_{{\kappa 1}+{\kappa 2}}^{p \times q} will be presented elsewhere. 相似文献
4.
D. S. Lubinsky 《Constructive Approximation》1988,4(1):321-339
Letf(z):=Σ j=0 ∞ a j z j , where aj ≠ 0,j large enough, and for someq ε C such that ¦q¦ $$q_j : = a_{j - 1} a_{j + 1} /a_j^2 \to q,j \to \infty .$$ Define for m,n = 0,1,2,..., the Toeplitz determinant $$D(m/n): = \det (a_{m - j + k} )_{j,k = 1}^n .$$ Given ? > 0, we show that form large enough, and for everyn = 1,2,3,..., $$(1 - \varepsilon )^n \leqslant \left| {{{D(m/n)} \mathord{\left/ {\vphantom {{D(m/n)} {\left\{ {a_m^n \mathop \Pi \limits_{j - 1}^{n - 1} (1 - q_m^j )^{n - j} } \right\}}}} \right. \kern-\nulldelimiterspace} {\left\{ {a_m^n \mathop \Pi \limits_{j - 1}^{n - 1} (1 - q_m^j )^{n - j} } \right\}}}} \right| \leqslant (1 + \varepsilon )^n .$$ We apply this to show that any sequence of Padé approximants {[m k /n k ]} 1 ∞ tof, withm k →∞ ask→ ∞, converges locally uniformly in C. In particular, the diagonal sequence {[n/n]} 1 ∞ converges throughout C. Further, under additional assumptions, we give sharper asymptotics forD(m/n). 相似文献
5.
We prove that the Banach space (?n=1¥lpn)lq(\bigoplus_{n=1}^{\infty}\ell_{p}^{n})_{\ell_{q}}, which is isomorphic to certain Besov spaces, has a greedy basis whenever 1≤p≤∞ and 1<q<∞. Furthermore, the Banach spaces (?n=1¥lpn)l1(\bigoplus_{n=1}^{\infty}\ell _{p}^{n})_{\ell_{1}}, with 1<p≤∞, and (?n=1¥lpn)c0(\bigoplus_{n=1}^{\infty}\ell_{p}^{n})_{c_{0}}, with 1≤p<∞, do not have a greedy basis. We prove as well that the space (?n=1¥lpn)lq(\bigoplus_{n=1}^{\infty}\ell _{p}^{n})_{\ell_{q}} has a 1-greedy basis if and only if 1≤p=q≤∞. 相似文献
6.
Tapani Matala-aho 《Constructive Approximation》2011,33(3):289-312
We shall present short proofs for type II (simultaneous) Hermite–Padé approximations of the generalized hypergeometric and
q-hypergeometric series
F(t)=?n=0¥\frac?k=0n-1P(k)?k=0n-1Q(k)tn, Fq(t)=?n=0¥\frac?k=0n-1P(qk)?k=0n-1Q(qk)tn,F(t)=\sum_{n=0}^{\infty}\frac{\prod_{k=0}^{n-1}P(k)}{\prod _{k=0}^{n-1}Q(k)}t^n,\qquad F_q(t)=\sum_{n=0}^{\infty}\frac{\prod_{k=0}^{n-1}P(q^k)}{\prod _{k=0}^{n-1}Q(q^k)}t^n, 相似文献
7.
We study Banach spaces of the form
We call such a space a p-space, p[1,), if for every k the space
is isomorphic to pk and the sequence (pk) strictly decreases to p. We examine the finite block representability of the spaces r in a p-space proving that it depends not only on p but also on the sequences (pk) and (nk). Assuming that i ni
1/q decreases to 0, where q is the conjugate exponent of p, we prove the existence of an asymptotic biorthogonal system in X and also that c
0 is finitely representable in X. Moreover we investigate the modified versions of p-spaces proving that, if nkm1/pkm-1/pkm-1 increases to infinity for a subsequence (nkm) , then 1 embeds into X. We also investigate complemented minimality for the class of spaces
where
is either a subsequence of the sequence of Schreier classes (
n)n N or a subsequence of (
n)n N. 相似文献
8.
In this paper, we consider the Schrödinger type operator ${H = (-\Delta _{\mathbb {H}}^n)^2 +V ^{2}}
9.
We study the properties of the polynomial operator pencil
10.
Zheng Yan LIN Sung Chul LEE 《数学学报(英文版)》2006,22(2):535-544
Let {Xn,n ≥ 0} be an AR(1) process. Let Q(n) be the rescaled range statistic, or the R/S statistic for {Xn} which is given by (max1≤k≤n(∑j=1^k(Xj - ^-Xn)) - min 1≤k≤n(∑j=1^k( Xj - ^Xn ))) /(n ^-1∑j=1^n(Xj -^-Xn)^2)^1/2 where ^-Xn = n^-1 ∑j=1^nXj. In this paper we show a law of iterated logarithm for rescaled range statistics Q(n) for AR(1) model. 相似文献
11.
E. M. E. ZAYED 《数学学报(英文版)》2005,21(4):733-752
The trace of the wave kernel μ(t) =∑ω=1^∞ exp(-itEω^1/2), where {Eω}ω^∞=1 are the eigenvalues of the negative Laplacian -△↓2 = -∑k^3=1 (δ/δxk)^2 in the (x^1, x^2, x^3)-space, is studied for a variety of bounded domains, where -∞ 〈 t 〈 ∞ and i= √-1. The dependence of μ (t) on the connectivity of bounded domains and the Dirichlet, Neumann and Robin boundary conditions are analyzed. Particular attention is given for a multi-connected vibrating membrane Ω in Ra surrounded by simply connected bounded domains Ω j with smooth bounding surfaces S j (j = 1,……, n), where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Si^* (i = 1 + kj-1,……, kj) of the bounding surfaces S j are considered, such that S j = Ui-1+kj-1^kj Si^*, where k0=0. The basic problem is to extract information on the geometry Ω by using the wave equation approach from a complete knowledge of its eigenvalues. Some geometrical quantities of Ω (e.g. the volume, the surface area, the mean curvuture and the Gaussian curvature) are determined from the asymptotic expansion ofexpansion of μ(t) for small │t│. 相似文献
12.
E.M.E.ZAYED 《数学学报(英文版)》2003,19(4):679-694
The asymptotic expansions of the trace of the heat kernel θ(t)=∑^∞v=1^exp(-tλv) for small positive t,where {λv} are the eigenvalues of the negative Laplacian -△n=-∑^ni=1(D/Dx^1)^2 in R^2(n=2 or 3),are studied for a general annular bounded domain Ω with a smooth inner boundary DΩ1 and a smooth outer boundary DΩ2,where a finite number of piecewise smooth Robin boundary conditions(D/Dnj γh)Ф=0 on the components Гj(j= 1,...,m) of (DΩ1 and on the components Гj (j=k 1,…,m) of of DΩ2 are considered such that DΩl=U^kj=lГj and DΩ2= U^m=k 1Гj and where the coefficients γj(j=1,...,m) are piecewise smooth positive functions. Some applications of θ(t) for an ideal gas enclosed in the general annular bounded domain Ω are given. Further results are also obtained. 相似文献
13.
Let X be a Banach space and let (ξj)j ≧ 1 be an i.i.d. sequence of symmetric random variables with finite moments of all orders. We prove that the following assertions
are equivalent:
14.
Stevan Pilipovi? Nenad Teofanov Joachim Toft 《Journal of Fourier Analysis and Applications》2011,17(3):374-407
Let ω,ω
0 be appropriate weight functions and q∈[1,∞]. We introduce the wave-front set, WFFLq(w)(f)\mathrm{WF}_{\mathcal{F}L^{q}_{(\omega)}}(f) of f ? S¢f\in \mathcal{S}' with respect to weighted Fourier Lebesgue space FLq(w)\mathcal{F}L^{q}_{(\omega )}. We prove that usual mapping properties for pseudo-differential operators Op (a) with symbols a in S(w0)r,0S^{(\omega _{0})}_{\rho ,0} hold for such wave-front sets. Especially we prove that
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