共查询到20条相似文献,搜索用时 15 毫秒
1.
《Applied and Computational Harmonic Analysis》2020,48(2):706-730
It was recently shown that on a large class of important Banach spaces there exist no linear methods which are able to approximate the Hilbert transform from samples of the given function. This implies that there is no linear algorithm for calculating the Hilbert transform which can be implemented on a digital computer and which converges for all functions from the corresponding Banach spaces. The present paper develops a much more general framework which also includes non-linear approximation methods. All algorithms within this framework have only to satisfy an axiom which guarantees the computability of the algorithm based on given samples of the function. The paper investigates whether there exists an algorithm within this general framework which converges to the Hilbert transform for all functions in these Banach spaces. It is shown that non-linear methods give actually no improvement over linear methods. Moreover, the paper discusses some consequences regarding the Turing computability of the Hilbert transform and the existence of computational bases in Banach spaces. 相似文献
2.
Nikos Yannakakis 《Set-Valued and Variational Analysis》2011,19(4):555-567
We show that if the conclusion of the well known Stampacchia Theorem on variational inequalities holds on a real Banach space
X, then X is isomorphic to a Hilbert space. Motivated by this, we obtain a relevant result concerning self-dual Banach spaces and investigate
some connections between properties of orthogonality relations, self-duality and Hilbert space structure. Moreover, we revisit
the notion of the cosine of a linear operator and show that it can be used to characterize real Banach spaces that are isomorphic
to a Hilbert space. Finally, we present some consequences of our results to quadratic forms and to evolution triples. 相似文献
3.
In this paper we investigate the deficiency indices theory and the selfad-joint and nonselfadjoint (dissipative, accumulative) extensions of the minimal symmetric direct sum Hamiltonian operators. In particular using the equivalence of the Lax-Phillips scattering matrix and the Sz.-Nagy-Foia¸s characteristic function, we prove that all root (eigen and associated) vectors of the maximal dissipative extensions of the minimal symmetric direct sum Hamiltonian operators are complete in the Hilbert spaces. 相似文献
4.
Banach空间中线性算子的Tseng度量广义逆 总被引:11,自引:2,他引:9
在 Banach空间中,利用 Banach几何方法及度量投影算子,将 E.H.Moors的学生,曾远荣(Y.Y. Tseng)在 Hilbert空间中为线性算子引入的 Tseng广义道,推广到 Banach空间,引入 Tseng度量广义逆(此时的 Tseng度量广义逆一般为齐性算子,而非线性算子),利用 Banach空间对偶映射与广义正交分解定理给出 Tseng度量广义道存在的充分必要条件.讨论了最大Tseng度量广义逆在最优化,控制论及微分方程不适定问题有着直接应用的一些基础性质. 相似文献
5.
P. Cannarsa G. Da Prato H. Frankowska 《Journal of Mathematical Analysis and Applications》2018,457(2):1173-1187
In a separable Banach space E, we study the invariance of a closed set K under the action of the evolution equation associated with a maximal dissipative linear operator A perturbed by a quasi-dissipative continuous term B. Using the distance to the closed set, we give a general necessary and sufficient condition for the invariance of K. Then, we apply our result to several examples of partial differential equations in Banach and Hilbert spaces. 相似文献
6.
Z.V. Kovarik 《Linear algebra and its applications》1975,12(3):223-229
A continuation of [6]. Gershgorin-type estimates for spectra in Banach spaces and Hilbert spaces are established when the set of perturbations of a given operator is a line segment, a linear image of the unit operator ball on a Hilbert space, and a ball of operators on a Banach space. 相似文献
7.
Virginia V. Jory 《Numerical Functional Analysis & Optimization》2013,34(4):477-492
Let H1 and H2 be Hilbert spaces and let B be a closed linear operator mapping a dense subset of H1 into H2. Several families of approximators which converge t o the orthogonal or Moore-Penrose generalized inverse B? are constructed. Additionally, the approximators are shown t o provide regularization operators for the equation Bx=y. Some of the results are extended to dissipative operators on reflexive and general Banach spaces. 相似文献
8.
We obtain continuous-time and discrete-time Lyapunov operator inequalities for the exponential stability of strongly continuous, one-parameter semigroups acting on Banach spaces. Thus we extend the classic result of Datko (1970) [2] from Hilbert spaces to Banach spaces. 相似文献
9.
黄乘明 《高等学校计算数学学报(英文版)》1996,(1)
In this paper, for general linear methods applied to strictly dissipative initial value problem in Hilbert spaces, we prove that algebraic stability implies B-convergence, which extends and improves the existing results on Runge-Kutta methods. Specializing our results for the case of multi-step Runge-Kutta methods, a series of B-convergence results are obtained. 相似文献
10.
Palle E. T. Jorgensen 《Central European Journal of Mathematics》2010,8(3):569-596
We study unbounded Hermitian operators with dense domain in Hilbert space. As is known, the obstruction for a Hermitian operator
to be selfadjoint or to have selfadjoint extensions is measured by a pair of deficiency indices, and associated deficiency
spaces; but in practical problems, the direct computation of these indices can be difficult. Instead, in this paper we identify
additional structures that throw light on the problem. We will attack the problem of computing deficiency spaces for a single
Hermitian operator with dense domain in a Hilbert space which occurs in a duality relation with a second Hermitian operator,
often in the same Hilbert space. 相似文献
11.
Simeon Reich 《Journal of Functional Analysis》1980,36(2):147-168
A special case of our main theorem, when combined with a known result of Brezis and Pazy, shows that in reflexive Banach spaces with a uniformly Gâteaux differentiable norm, resolvent consistency is equivalent to convergence for nonlinear contractive algorithms. (The linear case is due to Chernoff.) The proof uses ideas of Crandall, Liggett, and Baillon. Other applications of our theorem include results concerning the generation of nonlinear semigroups (e.g., a nonlinear Hille-Yosida theorem for “nice” Banach spaces that includes the familiar Hilbert space result), the geometry of Banach spaces, extensions of accretive operators, invariance criteria, and the asymptotic behavior of nonlinear semigroups and resolvents. The equivalence between resolvent consistency and convergence for nonlinear contractive algorithms seems to be new even in Hilbert space. Our nonlinear Hille-Yosida theorem is the first of its kind outside Hilbert space. It establishes a biunique correspondence between m-accretive operators and semigroups on nonexpansive retracts of “nice” Banach spaces and provides affirmative answers to two questions of Kato. 相似文献
12.
This paper focuses on boundedness and closedness of linear relations, which include both single-valued and multi-valued linear operators. A new (single-valued) linear operator induced by a linear relation is introduced, and its relationships with other two important induced linear operators are established. Several characterizations for closedness, closability, bundedness, relative boundedness and boundedness from below (above) of linear relations are given in terms of their induced linear operators. In particular, the closed graph theorem for linear relations in Banach spaces is completed, and stability of closedness of linear relations under bounded and relatively bounded perturbations is studied. The results obtained in the present paper generalize the corresponding results for single-valued linear operators to multi-valued linear operators, and some improve or relax certain assumptions of the related existing results. 相似文献
13.
In this paper we discuss stability problems for a class of discrete-time evolution operators generated by linear positive operators acting on certain ordered Banach spaces. Our approach is based upon a new representation result that links a positive operator with the adjoint operator of its restriction to a Hilbert subspace formed by sequences of Hilbert–Schmidt operators. This class includes the evolution operators involved in stability and optimal control problems for linear discrete-time stochastic systems. The inclusion is strict because, following the results of Choi, we have proved that there are positive operators on spaces of linear, bounded and self-adjoint operators which have not the representation that characterize the completely positive operators. As applications, we introduce a new concept of weak-detectability for pairs of positive operators, which we use to derive sufficient conditions for the existence of global and stabilizing solutions for a class of generalized discrete-time Riccati equations. Finally, assuming weak-detectability conditions and using the method of Lyapunov equations we derive a new stability criterion for positive evolution operators. 相似文献
14.
George Costakis Antonios Manoussos Ioannis Parissis 《Complex Analysis and Operator Theory》2014,8(8):1601-1643
We study the notion of recurrence and some of its variations for linear operators acting on Banach spaces. We characterize recurrence for several classes of linear operators such as weighted shifts, composition operators and multiplication operators on classical Banach spaces. We show that on separable complex Hilbert spaces the study of recurrent operators reduces, in many cases, to the study of unitary operators. Finally, we study the notion of product recurrence and state some relevant open questions. 相似文献
15.
Tepper L. Gill Sudeshna Basu Woodford W. Zachary V. Steadman 《Proceedings of the American Mathematical Society》2004,132(5):1429-1434
In this paper we show that a result of Gross and Kuelbs, used to study Gaussian measures on Banach spaces, makes it possible to construct an adjoint for operators on separable Banach spaces. This result is used to extend well-known theorems of von Neumann and Lax. We also partially solve an open problem on the existence of a Markushevich basis with unit norm and prove that all closed densely defined linear operators on a separable Banach space can be approximated by bounded operators. This last result extends a theorem of Kaufman for Hilbert spaces and allows us to define a new metric for closed densely defined linear operators on Banach spaces. As an application, we obtain a generalization of the Yosida approximator for semigroups of operators.
16.
§1.DefinitionsandNotationsLetEbearealBanachspace,andletx,y∈E.Thedirectionalderivative[x,y]-ofxandyisdefinedby(formoredetailss... 相似文献
17.
We present sufficient conditions for the linear asymptotic equilibrium of linear differential equations in Hilbert and Banach spaces. The results obtained are applied to studying the asymptotic equivalence of two linear differential equations. 相似文献
18.
A. R. Mirotin 《Mathematical Notes》1996,64(3):364-370
The functional calculus of several commuting dissipative elements of a complex Banach algebra with identity, first introduced in the preceding work by the author, is developed. Uniqueness, continuity, and stability theorems, composite function theorems, and a formula for the resolvent are established. Applications to the theory of sectorial operators in Hilbert space are given. 相似文献
19.
《Journal of Mathematical Analysis and Applications》1987,123(2):559-563
Some existence and uniqueness theorems of optimal nonparametric function estimation for Hilbert spaces are developed for Banach spaces. In particular, the results for reflexive Banach spaces parallel those for Hilbert spaces. 相似文献
20.
E. Pinto de Moura J.C. Robinson 《Journal of Mathematical Analysis and Applications》2010,368(1):254-262
This paper focuses on the regularity of linear embeddings of finite-dimensional subsets of Hilbert and Banach spaces into Euclidean spaces. We study orthogonal sequences in a Hilbert space H, whose elements tend to zero, and similar sequences in the space c0 of null sequences. The examples studied show that the results due to Hunt and Kaloshin (Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces, Nonlinearity 12 (1999) 1263-1275) and Robinson (Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces, Nonlinearity 22 (2009) 711-728) for subsets of Hilbert and Banach spaces with finite box-counting dimension are asymptotically sharp. An analogous argument allows us to obtain a lower bound for the power of the logarithmic correction term in an embedding theorem proved by Olson and Robinson (Almost bi-Lipschitz embeddings and almost homogeneous sets, Trans. Amer. Math. Soc. 362 (1) (2010) 145-168) for subsets X of Hilbert spaces when X−X has finite Assouad dimension. 相似文献