共查询到20条相似文献,搜索用时 31 毫秒
1.
Lu-Chuan Zeng 《Journal of Mathematical Analysis and Applications》2002,270(2):1-331
Let E be an arbitrary real Banach space and T :E→E be a Lipschitz continuous accretive operator. Under the lack of the assumption limn→∞αn=limn→∞βn=0, we prove that the Ishikawa iterative sequence with errors converges strongly to the unique solution of the equation x+Tx=f. Moreover, this result provides a convergence rate estimate for some special cases of such a sequence. Utilizing this result, we imply that if T :E→E is a Lipschitz continuous strongly accretive operator then the Ishikawa iterative sequence with errors converges strongly to the unique solution of the equation Tx=f. Our results improve, generalize and unify the ones of Liu, Chidume and Osilike, and to some extent, of Reich. 相似文献
2.
An extension of the Erdős–Ginzburg–Ziv Theorem to hypergraphs 总被引:1,自引:0,他引:1
David J. Grynkiewicz 《European Journal of Combinatorics》2005,26(8):1154-1176
An n-set partition of a sequence S is a collection of n nonempty subsequences of S, pairwise disjoint as sequences, such that every term of S belongs to exactly one of the subsequences, and the terms in each subsequence are all distinct with the result that they can be considered as sets. For a sequence S, subsequence S′, and set T, |T∩S| denotes the number of terms x of S with xT, and |S| denotes the length of S, and SS′ denotes the subsequence of S obtained by deleting all terms in S′. We first prove the following two additive number theory results.(1) Let S be a finite sequence of elements from an abelian group G. If S has an n-set partition, A=A1,…,An, such that then there exists a subsequence S′ of S, with length |S′|≤max{|S|−n+1,2n}, and with an n-set partition, , such that . Furthermore, if ||Ai|−|Aj||≤1 for all i and j, or if |Ai|≥3 for all i, then .(2) Let S be a sequence of elements from a finite abelian group G of order m, and suppose there exist a,bG such that . If |S|≥2m−1, then there exists an m-term zero-sum subsequence S′ of S with or .Let be a connected, finite m-uniform hypergraph, and be the least integer n such that for every 2-coloring (coloring with the elements of the cyclic group ) of the vertices of the complete m-uniform hypergraph , there exists a subhypergraph isomorphic to such that every edge in is monochromatic (such that for every edge e in the sum of the colors on e is zero). As a corollary to the above theorems, we show that if every subhypergraph of contains an edge with at least half of its vertices monovalent in , or if consists of two intersecting edges, then . This extends the Erdős–Ginzburg–Ziv Theorem, which is the case when is a single edge. 相似文献
3.
Let Bn( f,q;x), n=1,2,… be q-Bernstein polynomials of a function f : [0,1]→C. The polynomials Bn( f,1;x) are classical Bernstein polynomials. For q≠1 the properties of q-Bernstein polynomials differ essentially from those in the classical case. This paper deals with approximating properties of q-Bernstein polynomials in the case q>1 with respect to both n and q. Some estimates on the rate of convergence are given. In particular, it is proved that for a function f analytic in {z: |z|<q+} the rate of convergence of {Bn( f,q;x)} to f(x) in the norm of C[0,1] has the order q−n (versus 1/n for the classical Bernstein polynomials). Also iterates of q-Bernstein polynomials {Bnjn( f,q;x)}, where both n→∞ and jn→∞, are studied. It is shown that for q(0,1) the asymptotic behavior of such iterates is quite different from the classical case. In particular, the limit does not depend on the rate of jn→∞. 相似文献
4.
Vasiliy A. Prokhorov 《Journal of Approximation Theory》2000,107(2):337
Let E be a compact set in the extended complex plane C and let f be holomorphic on E. Denote by ρn the distance from f to the class of all rational functions of order at most n, measured with respect to the uniform norm on E. We obtain results characterizing the relationship between estimates of lim infn→∞ ρ1/nn and lim supn→∞ ρ1/nn. 相似文献
5.
Let be a domain with a Jordan boundary ∂G, consisting of l smooth curves Γj, such that {zj}Γj-1∩Γj≠, j=1,…,l, where Γ0Γl. Denote by αjπ, 0<αj2, the angles at zj's between the curves Γj-1 and Γj, exterior with respect to G. Let Φ be a conformal mapping of the exterior of onto the exterior of the unit disk, normed by Φ′(∞)>0. We assume that there is a neighborhood U of , such that , wherez≠zj if αj1. Set gGsup{|g(z)|:zG}. Then we prove Theorem. Let and 0βr. If a function f is analytic in G and f(r)βG<+∞, then for each nlr there is an algebraic polynomial Pn of degree <n, such that 相似文献
6.
Dale Umbach 《Journal of multivariate analysis》1978,8(4):518-531
The behavior of the posterior for a large observation is considered. Two basic situations are discussed; location vectors and natural parameters.Let X = (X1, X2, …, Xn) be an observation from a multivariate exponential distribution with that natural parameter Θ = (Θ1, Θ2, …, Θn). Let θx* be the posterior mode. Sufficient conditions are presented for the distribution of Θ − θx* given X = x to converge to a multivariate normal with mean vector 0 as |x| tends to infinity. These same conditions imply that E(Θ | X = x) − θx* converges to the zero vector as |x| tends to infinity.The posterior for an observation X = (X1, X2, …, Xn is considered for a location vector Θ = (Θ1, Θ2, …, Θn) as x gets large along a path, γ, in Rn. Sufficient conditions are given for the distribution of γ(t) − Θ given X = γ(t) to converge in law as t → ∞. Slightly stronger conditions ensure that γ(t) − E(Θ | X = γ(t)) converges to the mean of the limiting distribution.These basic results about the posterior mean are extended to cover other estimators. Loss functions which are convex functions of absolute error are considered. Let δ be a Bayes estimator for a loss function of this type. Generally, if the distribution of Θ − E(Θ | X = γ(t)) given X = γ(t) converges in law to a symmetric distribution as t → ∞, it is shown that δ(γ(t)) − E(Θ | X = γ(t)) → 0 as t → ∞. 相似文献
7.
We study the asymptotic behavior of the maximal multiplicity μn = μn(λ) of the parts in a partition λ of the positive integer n, assuming that λ is chosen uniformly at random from the set of all such partitions. We prove that πμn/(6n)1/2 converges weakly to max jXj/j as n→∞, where X1, X2, … are independent and exponentially distributed random variables with common mean equal to 1.2000 Mathematics Subject Classification: Primary—05A17; Secondary—11P82, 60C05, 60F05 相似文献
8.
Wolfgang Gehlen 《Journal of Approximation Theory》1999,101(2):110
Let fC[−1, 1] be real-valued. We consider the sequence of strong unicity constants (γn(f))n induced by the polynomials of best uniform approximation of f. It is proved that lim infn→∞ γn(f)=0, whenever f is not a polynomial. 相似文献
9.
Transcendence measures and algebraic growth of entire functions 总被引:1,自引:1,他引:0
In this paper we obtain estimates for certain transcendence measures of an entire function f. Using these estimates, we prove Bernstein, doubling and Markov inequalities for a polynomial P(z,w) in ℂ2 along the graph of f. These inequalities provide, in turn, estimates for the number of zeros of the function P(z,f(z)) in the disk of radius r, in terms of the degree of P and of r.
Our estimates hold for arbitrary entire functions f of finite order, and for a subsequence {n
j
} of degrees of polynomials. But for special classes of functions, including the Riemann ζ-function, they hold for all degrees
and are asymptotically best possible. From this theory we derive lower estimates for a certain algebraic measure of a set
of values f(E), in terms of the size of the set E. 相似文献
10.
Let {vij; i, J = 1, 2, …} be a family of i.i.d. random variables with E(v114) = ∞. For positive integers p, n with p = p(n) and p/n → y > 0 as n → ∞, let Mn = (1/n) Vn VnT , where Vn = (vij)1 ≤ i ≤ p, 1 ≤ j ≤ n, and let λmax(n) denote the largest eigenvalue of Mn. It is shown that
a.s. This result verifies the boundedness of E(v114) to be the weakest condition known to assure the almost sure convergence of λmax(n) for a class of sample covariance matrices. 相似文献
11.
Let denote the set of continuous n×n matrices on an interval . We say that is a nontrivial k-involution if where ζ=e-2πi/k, d0+d1++dk-1=n, and with . We say that is R-symmetric if R(t)A(t)R-1(t)=A(t), , and we show that if A is R-symmetric then solving x′=A(t)x or x′=A(t)x+f(t) reduces to solving k independent dℓ×dℓ systems, 0ℓk-1. We consider the asymptotic behavior of the solutions in the case where . Finally, we sketch analogous results for linear systems of difference equations. 相似文献
12.
Summary We study the following nonlinear method of approximation by trigonometric polynomials in this paper. For a periodic function f we take as an approximant a trigonometric polynomial of the form Gm(f ) := ∑kЄΛ f^(k) e (i k,x), where ΛZd is a set of cardinality m containing the indices of the m biggest (in absolute value) Fourier coefficients f^ (k) of function f . Note that Gm(f ) gives the best m-term approximant in the L2-norm and, therefore, for each f ЄL2, ║f-Gm(f )║2→0 as m →∞. It is known from previous results that in the case of p ≠2 the condition f ЄLp does not guarantee the convergence ║f-Gm(f )║p→0 as m →∞.. We study the following question. What conditions (in addition to f ЄLp) provide the convergence ║f-Gm(f )║p→0 as m →∞? In our previous paper [10] in the case 2< p ≤∞ we have found necessary and sufficient conditions on a decreasing sequence {An}n=1∞ to guarantee the Lp-convergence of {Gm(f )} for all f ЄLp , satisfying an (f ) ≤An , where {an (f )} is a decreasing rearrangement of absolute values of the Fourier coefficients of f. In this paper we are looking for necessary and sufficient conditions on a sequence {M (m)} such that the conditions f ЄLp and ║GM(m)(f ) - Gm(f )║p →0 as m →∞ imply ║f - Gm(f )║p →0 as m →∞. We have found these conditions in the case when p is an even number or p = ∞. 相似文献
13.
The basic result of the paper states: Let F1, …, Fn, F1′,…, Fn′ have proportional hazard functions with λ1 ,…, λn , λ1′ ,…, λn′ as the constants of proportionality. Let X(1) ≤ … ≤ X(n) (X(1)′ ≤ … ≤ X(n)′) be the order statistics in a sample of size n from the heterogeneous populations {F1 ,…, Fn}({F1′ ,…, Fn′}). Then (λ1 ,…, λn) majorizes (λ1′ ,…, λn′) implies that (X(1) ,…, X(n)) is stochastically larger than (X(1)′ ,…, X(n)′). Earlier results stochastically comparing individual order statistics are shown to be special cases. Applications of the main result are made in the study of the robustness of standard estimates of the failure rate of the exponential distribution, when observations actually come from a set of heterogeneous exponential distributions. Further applications are made to the comparisons of linear combinations of Weibull random variables and of binomial random variables. 相似文献
14.
Krzysztof Przes
awski 《Journal of Approximation Theory》1996,85(3):288-296
It is shown that for each convex bodyARnthere exists a naturally defined family
AC(Sn−1) such that for everyg
A, and every convex functionf: R→Rthe mappingy∫Sn−1 f(g(x)−y, x) dσ(x) has a minimizer which belongs toA. As an application, approximation of convex bodies by balls with respect toLpmetrics is discussed. 相似文献
15.
It is well known by a classical result of Bourgain–Fremlin–Talagrand that if K is a pointwise compact set of Borel functions on a Polish space then given any cluster point f of a sequence (fn)nω in K one can extract a subsequence (fnk)kω converging to f. In the present work we prove that this extraction can be achieved in a “Borel way.” This will prove in particular that the notion of analytic subspace of a separable Rosenthal compacta is absolute and does not depend on the particular choice of a dense sequence. 相似文献
16.
M. P. H. Wolff 《Journal of Approximation Theory》2001,113(2):229
Let E be a Banach space over
and let the densely defined closed linear operator A:
(A)E→E be discretely approximated by the sequence ((An,
(An)))n
of operators An where each An is densely defined in the Banach space Fn. Let σa(A) be the approximate point spectrum of A and let σ(An) denote the -pseudospectrum of An. Generalizing our own result, we show that σa(A)lim inf σ(An)=n
∩kn σ(Ak) holds for every >0. We deduce that then for every compact set K
limn dist(σa(A)∩K, σa(An))=0 provided there exists M>0 such that (λ−An)−1M dist(λ, σ(An))−1 holds for every n and every λ in the resolvent set ρ(An) of An. We finally treat the problem under which conditions σa(A) can be approximated from below. More precisely we investigate the problem: Under which assumptions does ∩>0 ∩n
kn σ, a(Ak)σa(A) hold where σ, a(A) denotes the -approximate pseudospectrum? 相似文献
17.
Let f:
be a continuous, 2π-periodic function and for each n ε
let tn(f; ·) denote the trigonometric polynomial of degree n interpolating f in the points 2kπ/(2n + 1) (k = 0, ±1, …, ±n). It was shown by J. Marcinkiewicz that limn → ∞ ∝02π¦f(θ) − tn(f θ)¦p dθ = 0 for every p > 0. We consider Lagrange interpolation of non-periodic functions by entire functions of exponential type τ > 0 in the points kπ/τ (k = 0, ± 1, ± 2, …) and obtain a result analogous to that of Marcinkiewicz. 相似文献
18.
B. Feigin M. Jimbo M. Kashiwara T. Miwa E. Mukhin Y. Takeyama 《European Journal of Combinatorics》2004,25(8):1197
Let V(Λi) (resp., V(−Λj)) be a fundamental integrable highest (resp., lowest) weight module of
. The tensor product V(Λi)V(−Λj) is filtered by submodules
, n≥0, n≡i−j mod 2, where viV(Λi) is the highest vector and
is an extremal vector. We show that Fn/Fn+2 is isomorphic to the level 0 extremal weight module V(n(Λ1−Λ0)). Using this we give a functional realization of the completion of V(Λi)V(−Λj) by the filtration (Fn)n≥0. The subspace of V(Λi)V(−Λj) of
-weight m is mapped to a certain space of sequences (Pn,l)n≥0,n≡i−jmod2,n−2l=m, whose members Pn,l=Pn,l(X1,…,Xlz1,…,zn) are symmetric polynomials in Xa and symmetric Laurent polynomials in zk, with additional constraints. When the parameter q is specialized to
, this construction settles a conjecture which arose in the study of form factors in integrable field theory. 相似文献
19.
Ana María Suchanek 《Journal of multivariate analysis》1978,8(4):589-597
A remarkable theorem proved by Komlòs [4] states that if {fn} is a bounded sequence in L1(R), then there exists a subsequence {fnk} and f L1(R) such that fnk (as well as any further subsequence) converges Cesaro to f almost everywhere. A similar theorem due to Révész [6] states that if {fn} is a bounded sequence in L2(R), then there is a subsequence {fnk} and f L2(R) such that Σk=1∞ ak(fnk − f) converges a.e. whenever Σk=1∞ | ak |2 < ∞. In this paper, we generalize these two theorems to functions with values in a Hilbert space (Theorems 3.1 and 3.3). 相似文献
20.
We consider the class of primitive stochastic n×n matrices A, whose exponent is at least (n2−2n+2)/2+2. It is known that for such an A, the associated directed graph has cycles of just two different lengths, say k and j with k>j, and that there is an α between 0 and 1 such that the characteristic polynomial of A is λn−αλn−j−(1−α)λn−k. In this paper, we prove that for any mn, if α1/2, then Am+k−Am∞Am−1wT∞, where 1 is the all-ones vector and wT is the left-Perron vector for A, normalized so that wT1=1. We also prove that if jn/2, n31 and
, then Am+j−Am∞Am−1wT∞ for all sufficiently large m. Both of these results lead to lower bounds on the rate of convergence of the sequence Am. 相似文献