Showing papers on "Bounded function published in 1974"

A Characterization of Banach Spaces Containing l1

Abstract: It is proved that a Banach space contains a subspace isomorphic to l1 if (and only if) it has a bounded sequence with no weak-Cauchy subsequence. The proof yields that a sequence of subsets of a given set has a subsequence that is either convergent or Boolean independent.

The surface energy of a bounded electron gas

Abstract: An exact expression for the exchange and correlation energy of an inhomogeneous electron gas, as defined by Hohenberg, Kohn and Sham (1966), is derived. This expression is separated into exchange and correlation terms and a formula linking the surface exchange energy of a half space to the Kohn-Sham one electron potential follows without approximation. For an infinite barrier model, the local density approximation gives a surface exchange energy 50% greater than the exact value, a large and previously unsuspected error. An exact evaluation of the surface correlation energy is not feasible, but the authors argue that the dominant contribution, arising from the difference in zero point energy between bounded and unbounded systems, can be estimated using a simple model. Numerical results, not dependent on the introduction of arbitrary plasmon wavevector cutoffs, give surface correlation energies larger than Lang and Kohn (1970), who work from a local formula, by a factor of six.

Interior estimates for Ritz-Galerkin methods

Abstract: Interior a priori error estimates in Sobolev norms are derived from interior RitzGalerkin equations which are common to a class of methods used in approximating solutions of second order elliptic boundary value problems. The estimates are valid for a large class of piecewise polynomial subspaces used in practice, which are defined on both uniform and nonuniform meshes. It is shown that the error in an interior domain 2 can be estimated with the best order of accuracy that is possible locally for the subspaces used plus the error in a weaker norm over a slightly larger domain which measures the effects from outside of the domain Q. Additional results are given in the case when the subspaces are defined on a uniform mesh. Applications to specific boundary value problems are given. 0. Introduction. There are presently many methods which are available for computing approximate solutions of elliptic boundary value problems which may be classified as Ritz-Galerkin type methods. Many of these methods differ from each other in some respects (for example, in how they treat the boundary conditions) but have much in common in that they have what may be called "interior Ritz-Galerkin equations" which are the same. Here we shall be concerned with finding interior estimates for the rate of convergence for such a class of methods which are consequences of these interior equations. Let us briefly describe, in a special case, the type of question we wish to consider. Let &2 be a bounded domain in RN with boundary M2 and consider, for simplicity, the problem of finding an approximate solution of a boundary value problem (0.1) \u =f in Q2, (0.2) Au= g on U2, where A is some boundary operator. Suppose now that we are given a one-parameter family of finite-dimensional subspaces Sh (0 < h < 1) of an appropriate Hilbert space in which u lies and that, for each h, we have computed an approximate solution Uh c Sh to u using some Ritz-Galerkin type method. Here we have in mind, for example, methods such as the "engineer's" finite element method [8], [22], the Aubin-Babuska penalty method [2], [4], the methods of Nitsche [12], [13] or the Received October 15, 1973. AMS (MOS) subject classifications (1970). Primary 65N30, 65N15. Copyright i 1974, American Mathematical Society

Bounded symmetric homogeneous domains in infinite dimensional spaces

Abstract: In this article, we exhibit a large class of Banach spaces whose open unit balls are bounded symmetric homogeneous domains. These Banach spaces, which we call J*-algebras, are linear spaces of operators mapping one Hilbert space into another and have a kind of Jordan tripte product structure. In particular, all Hilbert spaces and all B*--algebras are J*-algebras. Moreover, all four types of the classical Cartan domains and their infinite dimensional analogues are the open unit balls of J*-algebras, and the same holds for any finite or infinite product of these domains. Thus we have a setting in which a large number of bounded symmetric homogeneous domains may be studied simultaneously. A particular advantage of this setting is the interconnection which exists between function-theoretic problems and problems of functional analysis. This leads to a simplified discussion of both types of problems. We shall see that the open unit balls of J*-algebras are natural generalizations of the open unit disc of the complex plane. In fact, we give an explicit algebraic formula for Mobius transformations of these balls and show that the origin can be mapped to any desired operator in the ball with one of the Mobius transformations. An extremal form of the Schwarz lemma then leads immediately to the representation of each biholomorphic mapping between the open unit balls of two J*-algebras as a composition of a Mobius transformation and a linear isometry of one of the J*-algebras onto the other. Such linear isometries reduce to a multiplication by unitary operators for mappings in the identity component of the group of all biholomorphic mappings of the open unit ball of a C*-algebra with identity. However, in general, linear isometries of one J*-algebra onto another can be complicated. Still, using the mentioned Schwarz lemma and Mobius transformations, we show that all such linear isometries preserve the J*-structure. A consequence of these results is that the open unit balls of two J*-a{gebras are holomorphically equivalent if and only if the J*-algebras are isometrically isomorphic under a mapping preserving the J * structure. Another consequence is that the open unit ball of a J*-algebra is holomorphically equivalent to a product of balls if and only if the J*-algebra is isometrically isomorphic to a product of J*--algebras. The last result connects the factorization of domains with the factorization of J*-algebras and has a number of interesting applications. For example, using Cartan's classification of bounded symmetric domains in C n, we classify all J*-algebras of dimension less than 16. Moreover, we reduce the problem of classifying all finite dimensional J*-algebras to the problem of finding some J*-algebras whose open unit balls are holomorphically equivalent to the two exceptional Cartan domains in dimensions 16 and 27, respectively, when such J*-algebras exist. If there are such J*-atgebras in both cases, then every bounded symmetric

Spaces of Compact Operators

Abstract: In this paper we study the structure of the Banach space K(E, F) of all compact linear operators between two Banach spaces E and F. We study three distinct problems: weak compactness in K(E, F), subspaces isomorphic to l~ and complementation of K(E, F) in L(E, F), the space of bounded linear operators. In § 2 we derive a simple characterization of the weakly compact subsets of K(E, F) using a criterion of Grothendieck. This enables us to study reflexivity and weak sequential convergence. In § 3 a rather different problem is investigated from the same angle. Recent results of Tong [20] indicate that we should consider when K(E, F) may have a subspace isomorphic to l~. Although L(E, F) often has this property (e.g. take E = F =/2) it turns out that K(E, F) can only contain a copy of l~o if it inherits one from either E* or F. In § 4 these results are applied to improve the results obtained by Tong and also to approach the problem investigated by Tong and Wilken [21] of whether K(E, F) can be non-trivially complemented in L(E,F) (see also Thorp [19] and Arterburn and Whitley [2]). It should be pointed out that the general trend of this paper is to indicate that K(E, F) accurately reflects the structure of E and F, in the sense that it has few properties which are not directly inherited from E and F. It is also worth stressing that in general the theorems of the paper do not depend on the approximation property, which is now known to fail in some Banach spaces; the paper is constructed independently of the theory of tensor products. These results were presented at the Gregynog Colloquium in May

A Note on Complete Controllability and Stabilizability for Linear Control Systems in Hilbert Space

Abstract: We consider the linear control system $\dot x = Ax + Bu$. Here A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators $T(t)$ on a Hilbert space E, and B is a bounded linear operator from a Hilbert space H to E. We give sufficient conditions for the existence of a bounded linear operator K from E to H so that the control system with feedback control law $u(t) = Kx(t)$ has the zero solution asymptotically stable. In particular, we study the relationship between the concept of complete controllability and the existence of K.

Fixed point iterations using infinite matrices

Abstract: Let E be a closed, bounded, convex subset of a Banach space X, f: E → E. Consider the iteration scheme defined by , where A is a regular weighted mean matrix. For particular spaces X and functions f this iterative scheme converges to a fixed point of f. [5] extends or generalizes related work of Browder and Petryshyn, Dotson, Franks and Marzec, Johnson, Kannan, Mann, and Reinermann. This paper continues investigations begun in [6].

Axiomatisations of the average and a further generalisation of monotonic sequences

Abstract: by JOHN BIBBY(Received 2 November, 1972; revised 10 January, 1973)1. Introduction. A bounded monotonic sequence is convergent. This paper shows thata bounde sequencd e which is ^-monotonic (to be defined) also converges. Th prooe f generalisesone attributed to Professor R. A. Rankin by Copson [1]. The theorem requires two definitions:the first axiomatises the notio of " averagn e " and the second generalises th of e conceptmonotonicity.

On the uniform ergodic theorem. II

Abstract: Let \T s be a strongly continuous semigroup of bounded linear operators on a Banach space X, satisfying lim ^UrJ/i = 0. We prove the equivalence of the following conditions: (1) t~ f-T dr con- verges uniformly as t — °o. (2) The infinitesimal generator A has closed range. (3) lim \R exists in the uniform operator topology. In (6) the author has proved that for a bounded linear operator T on a Banach space X satisfying ||T"/«|| —* 0, N~ 2 ~ T" converges uniformly if and only if {I — T)X is closed. In this paper we obtain the analogous result for the continuous case, generalizing some results of Hille and Phillips (5, Theorem 18.8.4) without using the operational calculus devised by Hille. Abel convergence in the discrete case is treated in the appendix.

On the optimal control of a deterministic epidemic

Abstract: A control scheme for the immunisation of susceptibles in the KermackMcKendrick epidemic model for a closed population is proposed. The bounded control appears linearly in both dynamics and integral cost functionals and any optimal policies are of the "bang-bang" type. The approach uses Dynamic Programming and Pontryagin's Maximum Principle and allows one, for certain values of the cost and removal rates, to apply necessary and sufficient conditions for optimality and show that a one-switch candidate is the optimal control. In the remaining cases we are still able to show that an optimal control, if it exists, has at most one switch.

Linear operators and vector measures

Abstract: This paper is a continuation of a study of operators on function spaces initiated by the authors in [2] and [3]. The operators are studied in terms of their representing measures. In w 2 a characterization of compact operators is obtained using recent results of Brooks [1]. In w 3 factorization and characterization theorems for p-dominated and absolutely p-summing maps are given. The factorization results are motivated by Pietsch's work [5]. The general setting is as follows. Each of E and F is a B-space (= Banach space), H is a compact Hausdorff space and C(H, E) is the B-space (sup norm) of all continuous E-valued defined on H. We shall be interested in operators (= continuous linear transformations) L: C(H,E)--~F and representing measures m: Z---~B(E, F**), where B(E, F**) is the B-space of all operators from E into the bidual of E and E is the Borel o--algebra of subsets of H. A finitely additive set function m: Z--~B(E, F**) is called a representing measure if m has finite semivariation and Imz[ (= to ta l variation of the adjoint measure) is a regular Borel measure for each z~F~ (=closed unit ball in F*). The Riesz Representation Theorem in this setting asserts that to each operator L: C(H, E)---~F there corresponds a unique representing measure m: E--~B(E, F**) so that L(f)=~nfdm and IILII =~(H), where ffz denotes the semivariation; this association is denoted by L,,--.m. The reader may consult Brooks and Lewis [3] for a detailed discussion of this setting. In particular, CA will be the characteristic function of a set A, S(Z) will denote the scalar valued simple functions over Z, and U(Z) will be the uniform closure of S(E); the spaces SE(2~ ) and Ue(2~) are defined analogously for E-valued functions. The reader should note that if m~-~,L, then m(A)x=L**(r ). Also, we shall say that a representing measure m is strongly bounded (s-bounded) if rh(Ai)---~0 for a disjoint sequence (Ai)c S. The first named author acknowledges support of the National Science Foundation during the preparation of this paper.

Some Extensions of a Theorem of Hardy, Littlewood and Pólya and Their Applications

Abstract: In [6], by means of convex functions Φ :R→R, Hardy, Littlewood and Polya proved a theorem characterizing the strong spectral order relation for any two measurable functions which are defined on a finite interval and which they implicitly assumed to be essentially bounded (cf. [6, the approximation lemma on p. 150 and Theorem 9 on p. 151 of their paper]; see also L. Mirsky [10, pp. 328-329] and H. D. Brunk [1,Theorem A, p. 820]).

Commutators and Self-Adjointness of Hamiltonian Operators

Abstract: A time dependent approach to self-adjointness is presented and it is applied to quantum mechanical Hamiltonians which are not semi-bounded. Sufficient conditions are given for self-adjointness of Schrodinger and Dirac Hamiltonians with potentials which are unbounded at infinity. The method is the introduction of an auxiliary operatorN≧0 whose rate of change (commutator with the Hamiltonian) is bounded by a multiple ofN.

An existence proof for the Hartree-Fock time-dependent problem with bounded two-body interaction

Abstract: Using fixed point theorems for local contractions in Banach spaces, an existence and uniqueness proof for the Hartree-Fock time-dependent problem is given in the case of a finite Fermi system interacting via a bounded two-body potential. The existence proof for the “strong” solution of the evolution problem is obtained under suitable conditions on the initial state.

Sequential gradient-restoration algorithm for optimal control problems with nondifferential constraints

Abstract: This paper considers the numerical solution of optimal control problems involving a functionalI subject to differential constraints, nondifferential constraints, and terminal constraints. The problem is to find the statex(t), the controlu(t), and the parameter π so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy. The approach taken is a sequence of two-phase processes or cycles, composed of a gradient phase and a restoration phase. The gradient phase involves a single iteration and is designed to decrease the functional, while the constraints are satisfied to first order. The restoration phase involves one or several iterations and is designed to restore the constraints to a predetermined accuracy, while the norm of the variations of the control and the parameter is minimized. The principal property of the algorithm is that it produces a sequence of feasible suboptimal solutions: the functionsx(t),u(t), π obtained at the end of each cycle satisfy the constraints to a predetermined accuracy. Therefore, the functionals of any two elements of the sequence are comparable. The stepsize of the gradient phase is determined by a one-dimensional search on the augmented functionalJ, and the stepsize of the restoration phase by a one-dimensional search on the constraint errorP. If α g is the gradient stepsize and α r is the restoration stepsize, the gradient corrections are ofO(α g ) and the restoration corrections are ofO(α r α g 2). Therefore, for α g sufficiently small, the restoration phase preserves the descent property of the gradient phase: the functionalI at the end of any complete gradient-restoration cycle is smaller than the functionalI at the beginning of the cycle. To facilitate the numerical solution on digital computers, the actual time ϑ is replaced by the normalized timet, defined in such a way that the extremal arc has a normalized time length Δt=1. In this way, variable-time terminal conditions are transformed into fixed-time terminal conditions. The actual time τ at which the terminal boundary is reached is regarded to be a component of the parameter π being optimized. The present general formulation differs from that of Ref. 4 because of the inclusion of the nondifferential constraints to be satisfied everywhere over the interval 0 ≤t ≤ 1. Its importance lies in that (i) many optimization problems arise directly in the form considered here, (ii) problems involving state equality constraints can be reduced to the present scheme through suitable transformations, and (iii) problems involving inequality constraints can be reduced to the present scheme through suitable transformations. The latter statement applies, for instance, to the following situations: (a) problems with bounded control, (b) problems with bounded state, (c) problems with bounded time rate of change of the state, and (d) problems where some bound is imposed on an arbitrarily prescribed function of the parameter, the control, the state, and the time rate of change of the state. Numerical examples are presented for both the fixed-final-time case and the free-final-time case. These examples demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.

Minimal projections on hyperplanes in sequence spaces

Abstract: If Iz is a closed linear subspace in a Banach space X, then a projection of X onto Y is a bounded linear map P: X -> Y such that Py = y for all y E Y. If such a projection exists, then Y is said to be complemented in X. In this case, there is some interest in discovering whether there exist projections of minimal norm, and if so, what their properties are. Many applications of projections occur in numerical anMysis and approximation theory, for/~x can be regarded as an approximation to x in IT. The quality of this approximation relative to the best approximation is governed by the inequality

Some new results about Hammerstein equations

Abstract: A detailed discussion and a complete bibliography about equation (1) can be found in [3]. The new feature about the results we present here is the fact that we do not assume any coercivity for F. When Fis monotone and K maps L(Q) into Z,°°(£i), there is no growth restriction on F either (cf. Theorem 1). The monotonicity of F can be weakened when Kis compact (cf. Theorem 4). Also some of these results are valid for systems in the case where F is the gradient of a convex function (cf. Theorem 5). Assume (2) Kis a monotone hemicontinuous mapping from //(fi) into L°°(Q) which maps bounded sets into bounded sets, (3) f(x, r):CixR-^R is continuous and nondecreasing in r for a.e. x eQ, and is integrable in x for all r e R.

Variational principles for the invariant toroids of classical dynamics

Abstract: Kolmogorov (1957), Arnol'd (1973) and Moser (1962) proved that invariant toroids of N dimensions occupy a finite volume of the 2N-dimensional phase space of nearly integrable bounded systems of N degrees of freedom. Variational principles are stated for such invariant toroids.

An implicit sampling theorem for bounded bandlimited functions

Abstract: A rigorous proof of the 'strong bias tone' scheme is embodied in the implicit sampling theorem. The representation of signals that are sample functions of possible nonstationary random processes being of principal interest, the proof could not directly invoke results from classical analysis, which depend on the existence of the Fourier transform of the function under consideration; rather, it is based on Zakai's (1965) theorem on the series expansion of functions, band-limited under a suitably extended definition. A practical circuit that restores an approximate version of the signal from its sine-wave-crossings is presented and possible improvements to it are discussed.

Pseudo-harmonic interpolation on convex domains*

Abstract: Practical interpolation schemes for functions of more than one variable need not be finite dimensional as in the case of univariate functions The so-called “blending function methods” are one example of such transfinite schemes Another is the method of “pseudo-harmonic interpolation” described in this paper For any bounded, convex planar domain $\mathcal{D}$, we show how to construct a bivariate function U which interpolates to arbitrary Dirichlet boundary conditions on the perimeter of $\mathcal{D}$ The interpolant $U(Q)$ satisfies the familar “maximum principle” and, in the case in which the domain $\mathcal{D}$ is circular, is actually a harmonic function in $\mathcal{D}$ Higher order schemes are described in which the function $U(Q)$ interpolates to specified normal derivatives on the boundary of $\mathcal{D}$ as well as to the Dirichlet boundary conditions The interpolation schemes are also viewed as idempotent linear operators (ie, projectors) on families of continuous functions on $\overlin

Reversal-Bounded Acceptors and Intersections of Linear Languages

Abstract: A Turing machine whose behavior is restricted so that each read-write head can change its direction only a bounded number of times is reversal-bounded. Here we consider nondeterministic multitape acceptors which are both reversal-bounded and also operate in linear time. Our main result shows that such an acceptor need have only three pushdown stores as auxiliary storage, each pushdown store need make only one reversal, and the acceptor can operate in real time.

Approximations to stochastic programs with complete fixed recourse

Abstract: The probability distribution of the data entering a recourse problem is replaced by finite discrete distributions. It is proved that the convergence of the objective functions of the approximating problems to that one of the original problem can be achieved by choosing the discrete distributions in quite a natural way. For bounded feasible sets this implies the convergence of the optimal values. Finally some error bounds are derived.

The Radon-Nikodym property and dentable sets in Banach spaces

Abstract: In order to prove a Radon-Nikodym theorem for the Bochner integral, Rieffel [5] introduced the class of "dentable" subsets of Banach spaces. Maynard [31 later introduced the strictly larger class of "s-dentable" sets, and extended Rieffel's result to show that a Banach space has the Radon-Nikodym property if and only if every bounded nonempty subset of E is s-dentable. He left open, however, the question as to whether, in a space with the Radon-Nikodym property, every bounded nonenipty set is dentable. In the present note we give an elementary construction which shows this question has an affirmative answer. Definitions. A Banach space E has the Radon-Nikodym property if for each totally finite positive measure space (X, ', jt) and each E-valued, tcontinuous measure m on E with lmI(X) 0, there exists x e A such that x C cl co (A\S,(x)). [Here co B denotes the convex hull of B, cl co B is its closure and SE(x) = ty e E: llx y|| 0 there exists x e A such that x ' s(A\SE(x)). [Her s(B)---W i .Xx X. > 0? --.A= 1 Ixi C B}.] A point x E A is a denting [sdenting] point if for all E > 0, x V' cl co(A\S,(x)) [x v s(A\SE(x))]. Dentable sets are s-dentable, and Maynard has given an example of a bounded set which is s-dentable but not dentable. Rieffel has shown that if A is not dentable, then neither is clco A. The analogous assertion fails for s-dentability: The closed unit ball of C([O, 1]) is not dentable [5], but the constant ly 1 function is an s-denting point. By Lemma 2 below, its interior is not sdentable. Lemma 1. A subset A of E is not dentable if and only if there exists Received by the editors July 31, 1973. AMS (MOS) subject classifications (1970). Primary 46G05; Secondary 28A45.

Left centralizers of an *-algebra

Abstract: An explicit characterization is given of the left centralizers of a proper H*-algebra A. Each left centralizer is seen to correspond to a bounded family of bounded operators, where each operator acts on a Hilbert space associated with a minimalclosed two-sided ideal of A. Introduction. Let A be a semisimple Banach algebra. As in [3], we call a linear operator T on A a left centralizer if T(xy) = T(x)y, x, y E A. In this note we give an explicit characterization of the left centralizers on A when A is a proper H*-algebra. Centralizers on H*-algebras have been considered in [1], [6], and [9]. The same characterization holds when A is a dual B*-algebra, and has been given by Malviya and Tomiuk in [7]. Our proof is similar to that in [7]. We include most details for the sake of completeness. Use will be made of the structure theory of H*-algebras (see e.g. [8]), which we shall review here briefly, after introducing some notation. Given a family of Banach algebras, {Aylyer and numbers ky> 1, we denote by IP({Ay, ky}), ?