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1.
This paper studies the Drazin inverse for perturbed matrices. For that, given a square matrix A, we consider and characterize the class of matrices B with index s such that , and , where and denote the null space and the range space of a matrix A, respectively, and AD denote the Drazin inverse of A. Then, we provide explicit representations for BD and BBD, and upper bounds for the relative error BD-AD/AD and the error BBD-AAD. A numerical example illustrates that the obtained bounds are better than others given in the literature.  相似文献   

2.
The continuity conditions at the endpoints of interpolation theorems, TaBjMj aAj for j=0, 1, can be written with the help of the approximation functional: E(tTaB1B0)LM0 aA0 and E(tTaB0B1)LM1 aA1. As a special case of the results we present here we show that in the hypotheses of the interpolation theorem the L norms can be replaced by BMO( +) norms. This leads to a strong version of the Stein-Weiss theorem on interpolation with change of measure. Another application of our results is that the condition fL0, i.e., f*L, where f*(γ)=μ{|f|>γ} is the distribution function of f, can be replaced in interpolation with L(pq) spaces by the weaker f*BMO( +).  相似文献   

3.
4.
LetXbe a Banach space. GivenMa subspace ofXwe denote withPMthe metric projection ontoM. We defineπ(X) sup{PMMa proximinal subspace ofX}. In this paper we give a bound forπ(X). In particular, whenX=Lp, we obtain the inequality PM2|2/p−1|, for every subspaceMofLp. We also show thatπ(X)=π(X*).  相似文献   

5.
In this paper, we consider the semilinear initial value problem associated with an operator A whose spectrum lies in a sector of the complex plane and whose resolvent satisfies (zA)−1M|z|γ for some −1<γ<0 and all z outside the sector. The properties of existence and uniqueness of global mild solutions and continuous dependence on the initial data are investigated.  相似文献   

6.
In this paper we consider the trigonometric Fourier series with the β-general monotone coefficients. Necessary and sufficient conditions of L1-convergence for such a series, that is fSn=o(1), are obtained in terms of coefficients.  相似文献   

7.
For n1, let {xjn}nj=1 be n distinct points in a compact set K and letLn[·] denote the corresponding Lagrange interpolation operator. Let v be a suitably restricted function on K. What conditions on the array {xjn}1jnn1 ensure the existence of p>0 such that limn→∞ (fLn[f]) vLp(K)=0 for very continuous fK→ ? We show that it is necessary and sufficient that there exists r>0 with supn1 πnvLr(K) ∑nj=1 (1/|πn| (xjn))<∞. Here for n1, πn is a polynomial of degree n having {xjn}nj=1 as zeros. The necessity of this condition is due to Ying Guang Shi.  相似文献   

8.
In [A. Biró, V.T. Sós, Strong characterizing sequences in simultaneous Diophantine approximation, J. Number Theory 99 (2003) 405–414] we proved that if Γ is a subgroup of the torus R/Z generated by finitely many independent irrationals, then there is an infinite subset AZ which characterizes Γ in the sense that for γR/Z we have ∑aAaγ<∞ if and only if γΓ. Here we consider a general compact metrizable Abelian group G instead of R/Z, and we characterize its finitely generated free subgroups Γ by subsets AG*, where G* is the Pontriagin dual of G. For this case we prove stronger forms of the analogue of the theorem of the above mentioned work, and we find necessary and sufficient conditions for a kind of strengthening of this statement to be true.  相似文献   

9.
The best possible constant An in an inequality of Markov type
, where ·[0, ∞) denotes the sup-norm on the half real line [0, ∞) and pn is an arbitrary polynomial of degree at most n, is determined in terms of the weighted Chebyshev polynomials associated with the Laguerre weight ex on [0, ∞).  相似文献   

10.
It is known that shape preserving approximation has lower rates than unconstrained approximation. This is especially true for copositive and intertwining approximations. ForfLp, 1p<∞, the former only has rateω(fn−1)p, and the latter cannot even be bounded byC fp. In this paper, we discuss various ways to relax the restrictions in these approximations and conclude that the most sensible way is the so-calledalmostcopositive/intertwining approximation in which one relaxes the restriction on the approximants in a neighborhood of radiusΔn(yj) of each sign changeyj.  相似文献   

11.
For the weight function (1−x2)μ−1/2 on the unit ball, a closed formula of the reproducing kernel is modified to include the case −1/2<μ<0. The new formula is used to study the orthogonal projection of the weighted L2 space onto the space of polynomials of degree at most n, and it is proved that the uniform norm of the projection operator has the growth rate of n(d−1)/2 for μ<0, which is the smallest possible growth rate among all projections, while the rate for μ0 is nμ+(d−1)/2.  相似文献   

12.
Consider the problem of estimating the mean vector θ of a random variable X in , with a spherically symmetric density f(xθ2), under loss δθ2. We give an increasing sequence of bounds on the shrinkage constant of Stein-type estimators depending on properties of f(t) that unify and extend several classical bounds from the literature. The basic way to view the conditions on f(t) is that the distribution of X arises as the projection of a spherically symmetric vector (X,U) in . A second way is that f(t) satisfies (−1)jf(j)(t)≥0 for 0≤j and that (−1)f()(t) is non-increasing where k=2(+1). The case =0 (k=2) corresponds to unimodality, while the case =k= corresponds to complete monotonicity of f(t) (or equivalently that f(xθ2) is a scale mixture of normals). The bounds on the minimax shrinkage constant in this paper agree with the classical bounds in the literature for the case of spherical symmetry, spherical symmetry and unimodality, and scale mixtures of normals. However, they extend these bounds to an increasing sequence (in k or ) of minimax bounds.  相似文献   

13.
We show that a sequentially (τ)-complete topological vector lattice Xτ is isomorphic to some L1(μ), if and only if the positive cone can be written as X+ = +B for some convex, (τ)-bounded, and (τ)-closed set B X+ {0}. The same result holds under weaker hypotheses, namely the Riesz decomposition property for X (not assumed to be a vector lattice) and the monotonic σ-completeness (monotonic Cauchy sequences converge). The isometric part of the main result implies the well-known representation theorem of Kakutani for (AL)-spaces. As an application we show that on a normed space Y of infinite dimension, the “ball-generated” ordering induced by the cone Y+ = + (for u >) cannot have the Riesz decomposition property. A second application deals with a pointwise ordering on a space of multivariate polynomials.  相似文献   

14.
In [O. Dragičević, A. Volberg, Sharp estimate of the Ahlfors–Beurling operator via averaging martingale transforms, Michigan Math. J. 51 (2) (2003) 415–435] the Ahlfors–Beurling operator T was represented as an average of two-dimensional martingale transforms. The same result can be proven for powers Tn. Motivated by [T. Iwaniec, G. Martin, Riesz transforms and related singular integrals, J. Reine Angew. Math. 473 (1996) 25–57], we deduce from here that Tnp are bounded from above by Cnp*, . We further improve this estimate to obtain the optimal behaviour of the Lp norms in question.  相似文献   

15.
Suppose that G is a graph with n vertices and m edges, and let μ be the spectral radius of its adjacency matrix.Recently we showed that if G has no 4-cycle, then μ2-μn-1, with equality if and only if G is the friendship graph.Here we prove that if m9 and G has no 4-cycle, then μ2m, with equality if G is a star. For 4m8 this assertion fails.  相似文献   

16.
We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e. spaces X such that the norm equality Id+T2=1+T2 holds for every bounded linear operator . This answers in the positive Question 4.11 of [V. Kadets, M. Martín, J. Merí, Norm equalities for operators on Banach spaces, Indiana Univ. Math. J. 56 (2007) 2385–2411]. More concretely, we show that this is the case of some C(K) spaces with few operators constructed in [P. Koszmider, Banach spaces of continuous functions with few operators, Math. Ann. 330 (2004) 151–183] and [G. Plebanek, A construction of a Banach space C(K) with few operators, Topology Appl. 143 (2004) 217–239]. We also construct compact spaces K1 and K2 such that C(K1) and C(K2) are extremely non-complex, C(K1) contains a complemented copy of C(2ω) and C(K2) contains a (1-complemented) isometric copy of .  相似文献   

17.
We construct a broad class of generalized Bayes minimax estimators of the mean of a multivariate normal distribution with covariance equal to σ2Ip, with σ2 unknown, and under the invariant loss δ(X)−θ2/σ2. Examples that illustrate the theory are given. Most notably it is shown that a hierarchical version of the multivariate Student-t prior yields a Bayes minimax estimate.  相似文献   

18.
For a compact convex set the well-known general Markov inequality holds asserting that a polynomial p of degree n must have pc(K)n2p. On the other hand for polynomials in general, p can be arbitrarily small as compared to p.The situation changes when we assume that the polynomials in question have all their zeroes in the convex set K. This was first investigated by Turán, who showed the lower bounds p(n/2)p for the unit disk D and for the unit interval I[-1,1]. Although partial results provided general lower estimates of order , as well as certain classes of domains with lower bounds of order n, it was not clear what order of magnitude the general convex domains may admit here.Here we show that for all bounded and convex domains K with nonempty interior and polynomials p with all their zeroes lying in K pc(K)np holds true, while pC(K)np occurs for any K. Actually, we determine c(K) and C(K) within a factor of absolute numerical constant.  相似文献   

19.
In this paper we investigate the existence of holey self-orthogonal Latin squares with a symmetric orthogonal mate of type 2nu1 (HSOLSSOM(2nu1)). For u2, necessary conditions for existence of such an HSOLSSOM are that u must be even and n3u/2+1. Xu Yunqing and Hu Yuwang have shown that these HSOLSSOMs exist whenever either (1) n9 and n3u/2+1 or (2) n263 and n2(u-2). In this paper we show that in (1) the condition n9 can be extended to n30 and that in (2), the condition n263 can be improved to n4, except possibly for 19 pairs (n,u), the largest of which is (53,28).  相似文献   

20.
Let Ak,k=0,1,2,…, be a sequence of real nonsingular n×n matrices which converge to a nonsingular matrix A. Suppose that A has exactly one positive eigenvalue λ and there exists a unique nonnegative vector u with properties Au=λu and u=1. Under further additional conditions on the spectrum of A, it is shown that if x0≠0 and the iterates
are nonnegative, then converges to u and converges to λ as k.  相似文献   

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