Finite-dimensional Projections Research Articles

On the finite-rank approximation of wave operators and the S matrix using Hermite expansion functions

For the quantum mechanical, nonrelativistic two-body system, wave operators can be approximated by exponentials of finite-rank operators obtained by finite-dimensional projections of the full and the asymptotic Hamiltonian onto a set of Hermite functions. This yields a weak approximation of the S matrix.

The structure of discrete-time finite-dimensional control of distributed parameter systems

In many applications, the distributed parameter nature of engineering systems cannot be ignored. This leads to state space descriptions of these systems on infinite-dimensional spaces. Yet, feedback controllers for engineering applications must be implemented with on-line digital computers; hence, the controllers must be discrete-time, finite-dimensional systems. The principal question is whether such a controller can achieve exponentially stable control of an infinite-dimensional system. The main result (Theorem 5) shows that, under some technical restrictions, all such stable controllers for linear distributed parameter systems (DPS) can do no better than feedback based on a finite-dimensional projection of the DPS state. Such stabilizing subspaces associated with the controller reveal the mathematical structure of the DPS control problem, but are not easy to calculate explicitly. The more natural approach is to select model-reducing subspaces, which approximate the open-loop DPS, and synthesize the controller from the reduced-order model. Stability bounds are presented (Theorem 7) and connections are made with the structural results (Theorem 5).

Foundation of the spectral approach in nonconservative problems of the theory of elastic stability

THEOREM I. Assume that the operator F generates in the Hilbert space X the C0-semigroup etF, and assume that o(F) is discrete and can be partitioned into the subsets { n}n=l to each of whicN corresponds a finite-dimensional spectral projection Pn (n = I, 2,...). We assume that the subspaces X n ~ Pn X (n = I, 2,...) have dimension not exceeding N and that they form a Riesz basis from subspaces in X [7]. Then we have the inequality

Stationarity on Finite Strings and Shift Register Sequences

Stationarity is a property of infinite sequences of random variables. An appropriate extension of this definition is made, to cover finite sequences. The set of finite stationary sequences is shown to be a convex set and its extreme points are related to shift register sequences (which are paths on a graph known as the shift net, or the de Bruijn graph). The set of finite stationary sequences as defined here is simply the set of finite dimensional projections of infinite stationary sequences.

Characterizing the autocorrelations of binary sequences

The class of autocorrelation functions of binary sequences is studied in terms of their finite-dimensional projections; that is, the vector ranges of finite sets of consecutive correlation coefficients. These are convex polyhedra whose extreme points can be generated by binary sequences of bounded periods. Algorithms are given for the calculation of the extreme points. Tables are presented for the extreme points and supporting hyperplanes of projections of dimensions up to five. Some conclusions about the general class of autocorrelations of binary sequences are drawn from these tables.

Finite dimensional projection constants

Finite dimensional projection constants

Point-spectrum of semibounded operator extensions

Let $\tilde H$ denote the Friedrichs extension of a given semibounded operator $H$ in a Hilbert space. Assume $\lambda I \leqslant H$, and $\lambda \in \sigma (\tilde H)$. If for a finite-dimensional projection $P$ in the Hubert space we have $I - P \leqslant$ Const. $(H - \lambda I)$, then it follows that $\lambda$ is an eigenvalue of $\tilde H$, and the corresponding eigenspace is contained in the range of $P$. Using this, together with the known order structure on the family of selfadjoint extensions, with given lower bound 0, of minus the Laplace-Beltrami operator, we establish the identity ${U_g}(1) = 1$ for all $g \in G$ for the following problem. $U$ is a unitary representation of a Lie group $G$, and acts on the Hilbert space ${L^2}(\Omega )$ for some Nikodym-domain $\Omega \subset G$. Moreover $U$ is obtained as a certain normalized integral for the left-$G$-in variant vector fields on $\Omega$, that is, for each such vector field $X$, the skew-adjoint operator $dU(X)$ is an extension of $X$ when regarded as a skew-symmetric operator in ${L^2}(\Omega )$ with domain $C_0^\infty (\Omega )$.

A decomposition theorem for finite-dimensional Hermitian projections

We investigate conditions for the existence of a decomposition of a Hermitian projector ρ into two Hermitian and time-reversal invariant operators ρ0 and χ under the form ρ=eiχ ρ0 e−iχ. Sufficient conditions are given and an explicit construction of a decomposition is performed when they are fulfilled. A stronger theorem of existence and uniqueness is studied.

Finite-dimensional projection operators of unit norm in the space C

Finite-dimensional projection operators of unit norm in the space C

Quasi-compactness and uniform ergodicity of positive operators

The following converse to the Yosida-Kakutani theorem is proved: IfT is a positive operator on a Banach lattice with ‖Tn‖/n → 0, thenT is quasi-compact if (and only if) the averages of its iterates converge uniformly to a finite-dimensional projection.

Compact Perturbations of Certain von Neumann Algebras

Let $\mathcal {E}$ be a sequence of mutually orthogonal, finite dimensional projections whose sum is the identity on a Hilbert space $\mathcal {H}$. If we denote the commutant of $\mathcal {E}$ by $\mathcal {D}(\mathcal {E})$ and the ideal of compact operators on $\mathcal {H}$ by $\mathcal {C}(\mathcal {H})$, then it is easily verified that $\mathcal {D}(\mathcal {E}) + \mathcal {C}(\mathcal {H}) = \{ T + K:T \in \mathcal {D}(\mathcal {E}),K \in \mathcal {C}(\mathcal {H})\}$ is a ${C^\ast }$-algebra. In this paper we classify all such algebras up to $^\ast$-isomorphism and characterize them by examining their relationship to certain quasidiagonal and quasitriangular algebras.

Compact perturbations of certain von Neumann algebras

Let E \mathcal {E} be a sequence of mutually orthogonal, finite dimensional projections whose sum is the identity on a Hilbert space H \mathcal {H} . If we denote the commutant of E \mathcal {E} by D ( E ) \mathcal {D}(\mathcal {E}) and the ideal of compact operators on H \mathcal {H} by C ( H ) \mathcal {C}(\mathcal {H}) , then it is easily verified that D ( E ) + C ( H ) = { T + K : T ∈ D ( E ) , K ∈ C ( H ) } \mathcal {D}(\mathcal {E}) + \mathcal {C}(\mathcal {H}) = \{ T + K:T \in \mathcal {D}(\mathcal {E}),K \in \mathcal {C}(\mathcal {H})\} is a C ∗ {C^\ast } -algebra. In this paper we classify all such algebras up to ∗ ^\ast -isomorphism and characterize them by examining their relationship to certain quasidiagonal and quasitriangular algebras.

Uniform Convergence of the Potential Function Algorithm

The identification problem of concern here is the estimation of a real function $f(x)$ by means of noisy observations $\{ (X_i ,f(X_i ) + \eta _i (X_i ))\} $ of its pairs, the $X_i $’s being chosen independently according to some fixed law P. The approach taken for estimation is the “potential function” method (its sources are referenced herein), to wit: Choose $f_0 $ arbitrarily and define the sequence $\{ f_n \} $ by the recursive relation $f_{n + 1} (x) = f_n (x) + \gamma _n (f(X_{n + 1} ) + \eta (X_{n + 1} ) - f_n (X_{n + 1} ))K(X_{n + 1} ,x)$, K being a positive symmetric kernel. From earlier publications it is known that under certain mild restrictions $E[\| {f_n - f} \|^2 ] \to 0$ in the $L_2 (p)$-norm. Rates of convergence have been obtained in the restrictive case that $K(x,y) = \sum_{i - 1}^N {\lambda _i^2 } \phi _i (x)\phi _i (y)$ and $f(x) \in {\operatorname{span}}\{ \phi _i ,1 \leqq i \leqq N\} $. The contribution of this paper is to prove that while no uniform bounds exist in the $L_2 (p)$-norm (we prove this) if $\{ \phi _i \} $ is an infinite set, we do have $E[\| {f - f_n } \|_k ^2 ] < C_n (\| f \|)$ for the norm $\| g \|_k^2 = \int {\int g (x)g(y)K(x,y)p(x)p(y)dxdy} $ and $\{ C_n (r)\} $ a sequence converging to 0 for each positive r. A final result concerns the rate at which increasing finite-dimensional projections of $f_n - f$ converge to 0 in the $L_2 (p)$-norm. From our methods it is seen that if $f \notin V = {\operatorname{span}}(\{ \phi _i \} )$, then $f_n $ converges in the mean to the projection of f on V.

Probability measures on distribution spaces and quantum field theoretical models

It is well known that there is a close connection between (Abelian) fields and probability measures on distribution space, or more generally, on infinite-dimensional vector spaces, and their associated random processes. We establish general criteria for mutual singularity of such measures and apply them to quantum fields. In particular, it is shown that a kind of generalized clustering implies singularity. Then a condition for singularity is given in terms of a natural metric introduced previously. It is used to show that the translate of a measure by a linear functional which is not continuous for this metric is singular to the untranslated measure. The results are applied to processes with independent values at each point, corresponding to the ultra-local model. It is shown that each such measuris singular to any of its translates although its finite-dimensional projections may be equivalent to Lebesgue measure.

The projection operator applied to gradient methods for solving optimal control problems with terminal state constraints

The projection operator usually associated with the steepest descent solution of constrained optimal control problems is developed by analogy to the familiar finite-dimensional projection matrix. The equivalence of two different derivations appearing in the literature is shown, and the geometric significance of various terms in the control variation equation of steepest descent is presented. A full derivation of the auxiliary differential equations used to implement the projection operator has not appeared previously bnt is given here for completeness. A simple parameter is introduced to provide both insight into the geometrie relationship of the performance and constraint contours and a measure of the convergence of the steepest descent algorithm. The application of the projection operator to other direct gradient search algorithms is discussed and illustrated by an application to the conjugate gradient method.

Some continuity properites of Brownian motion with the time parameter in Hilbert space

Levy introduced this process in [5, Chapter 8]. A bibliography of his later work is given in the second edition. The sample functions of this process are continuous when H is finite dimensional; however, when H is infinite dimensional, the sample functions are not only discontinuous but also unbounded on every sphere. The local irregularity of the sample functions is caused by the property of infinite-dimensional Hilbert space that a point can be approached from any one of infinitely many mutually orthogonal directions: a sample function has a negligible chance of being well behaved on all of the paths leading to a point. We ask: what are some sets upon which the sample functions are continuous? A simple example is a finitedimensional subspace, as we have already indicated; it is therefore natural to propose sets which are topologically almost in the sense that the mutual distances of their points are determined mostly by finite-dimensional projections of these points. Such a set is a cube. P. T. Strait proved the continuity of the sample functions on cubes whose edges have lengths 1, 2-1, 2-2,. .. [10]. Here we shall study the continuity of the sample functions on ellipsoids. Our results are: (i) A necessary condition for the continuity on an ellipsoid is that the lengths of the semiaxes converge to 0. (ii) The above condition is not sufficient: we can construct such an ellipsoid upon which the sample functions are unbounded. (iii) A sufficient condition for continuity is that the lengths of the semiaxes form a pth power summable sequence for some p < 1. This implies Strait's result