Showing papers on "Shift operator published in 1974"

Generalized Inverses in Reproducing Kernel Spaces: An Approach to Regularization of Linear Operator Equations

Abstract: In this paper a study of generalized inverses of linear operators in reproducing kernel Hilbert spaces (RKHS) is initiated. Explicit expressions for generalized inverses and minimal-norm solutions of linear operator equations in RKHS are obtained in several forms. The relation between the regularization operator of the equation $Af = g$ and the generalized inverse of the operator A in RKHS is demonstrated. In particular, it is shown that they are the same if the range of the operator is closed in an appropriate RKHS. Finally, properties of the regularized pseudosolutions in this setting are studied.

On the coupling operator method

Abstract: The coupling operator method in the general SCF theory is discussed in terms of the projection operator property of the density operator. The advantage of using the density operator is that one can put the arbitrariness of the general SCF orbitals in evidence. We showed how to put the fundamental condition for the optimum orbitals of the general SCF theory into a more general and useful form. The essential point of the coupling operator is how the variational conditions are included as the projections onto the intermanifolds. By using the arbitrariness of the manifold, we indicate how a modified SCF theory is developed, where by choosing an appropriate operator one can get any desired modified orbitals and their associated orbital energies under orbital transformations. The Appendix contains an extension of Koopmans' theorem as an application of the modified SCF theory. Finally the effective Hamiltonian is derived which is valid for almost all of the proposed SCF theory.

Regularization and approximation of linear operator equations in reproducing kernel spaces

Abstract: for a given y G Y if inf {\\Ax y II : x G V(A)} = \\Au y II. A pseudosolution of (1) for a given y G 7 is a least-squares solution of minimal norm. Equation (I) is well-posed relative to the spaces X, Y if for each y € Y9 (1) has a unique pseudosolution which depends continuously on y\ otherwise the equation is said to be ill-posed. One objective of this research is to show, when X and Y are L2spaces of square-integrable functions, that the topology of reproducing kernel Hilbert spaces (RKHS) is an appropriate topology for the regularization of ill-posed linear operator equations, and to initiate a study of generalized inverses of linear operators acting between two RKHS. A second objective is to provide an approach to optimal approximations of linear operator equations in the context of RKHS, and to demonstrate the relation between the regularization operator of the equation Af = g and the generalized inverse of A in an appropriate RKHS. (For some background on regularization methods see [3], [5], [9] ; for generalized inverses see, for example, [4].)

The inverse of a linear operator

Abstract: The result of the inverse of a linear operator on a vector is given by a recursion method which involves only three vectors at a time. The method is independent of any inner product.

The convergence of approximate methods for solving linear and non-linear operator equations

Abstract: The solutions of “approximate” operator equations A n x n = y n are taken as the approximate solutions of the operator equation Ax = y . The concept of the proper convergence of a sequence of operators is introduced, and used as a basis for proving the convergence of the approximate solutions. Convergence theorems are proved for linear equations, for non-linear equations with differentiable operators, and for non-linear equations with a linear principal, and completely continuous subordinate, part.

On the approximate solution of operator equations

Abstract: Prolongations and restrictions are used to derive error estimates when linear operator equations of the second kind are solved by discretisation methods.

The non-additive ligand field

Abstract: The semi-empirical ligand field is a perturbation operator whose consequences are taken to first order using a basis set ofl functions. Since the basis spans an irreducible representation of the 3-dimensional rotation-inversion groupR 3i it is useful to express the operator as a sum of components of irreducible tensor operators with respect to this group. IfR 3i is reduced with respect to the molecular subgroup the electronic factor of each term in the sum must be totally symmetrical within this group. This choice of operator leads to thecrystal field parameterization without implying an electrostatic model. Alternatively a shift operator withinl space may be chosen as the essential part of the perturbation operator. This leads to theligand field parameterization. Between the two parameterizations there exists a one to one relationship, whose coefficients are proportional to 3l symbols. This relationship is given together with a brief discussion of the reasons for the proposed parameterizations.

Passive Operator Networks

Abstract: The concept of an RLC network is introduced, where each R, L, and C-instead of being a positive number or matrix-is a positive bounded linear operator on a Hibert space H. Actually, this idea of a network of operators is generalized still further since in the frequency domain the branch impedances or admittances are only required to be certain kinds of operator-valued positive-real functions. Such operator networks can be used to study various types of infinite networks that are natural models for integrated and distributed networks. The procedure is to decompose a given infinite RLC network into infinite one-element-kind subnetworks in such a way that each subnetwork is characterized by either R, Ld/dt , or C^{-i} \int_{-\infty}^{t} {\cdots} dx , where now R, L, and C are positive invertible operators on Hilbert's coordinate space l_{2} . Some fairly general conditions on the subnetworks are presented, which insure the validity of the operator representations. At this point, essential use is made of some recently published results by Flanders concerning the uniqueness of the behavior of infinite networks. In the general case, where l_{2} is replaced by H, it is shown that the fundamental theorem of electrical networks continues to hold, so that H-valued voltage and current sources produce unique current distributions throughout the network. Moreover, the driving-point impedances of such networks are shown to be operator-valued forms of positive-real functions. A more special characterization for the driving-point impedances of RL and RC-type operator networks is also established. Mutual coupling can also be taken into account.

On the modulus of continuity of an operator of best approximation

Abstract: In this paper we characterize spaces with an operator of best approximation uniformly continuous on a class of subspaces.

On the unitary foldy-wouthuysen operator for particles of arbitrary spin

Abstract: A generalized Foldy-Wouthuysen operator which remains unitary and at the same time related to the Lorentz transformation has been obtained here for particles of arbitrary spin. The operator has a simple exponential form and is obtained as a product of two operatorsS andL. While the operatorL is obtained from the Lorentz transformation, the operatorS is constructed such thatSL becomes unitary. The Hamiltonian, obtained here by making a similarity transformation of the rest frame Hamiltonian with this generalized operator, is found to be exactly the same as that obtained by Weaver, Hammer and Good and by Mathews independently. A similar generalized unitary operator to get at the extreme relativistic representation is also given here.

Operator Valued Representations of Function Algebras

Abstract: The purpose of the present expository paper is to discuss basic ideas of operator representations of function algebras. Such representations originated from dilation theory. The relationship between dilations and representations is described in the first section. The second section deals partly with a dilation free approach to representations.

Structure of the point spectrum of a linear operator

Abstract: In order that a set of complex numbers be the point spectrum of some linear operator in a separable Hilbert space it is necessary and sufficient that it be a set of type Fσ.

O(3) shift operators and the group SL(3,R)

Abstract: A preliminary analysis of irreducible unitary representations of SL(3,R) is given using O(3) shift operator techniques similar to those used for treating SU(3) in an O(3) basis. A full analysis is given of the Delta l=2 representations, for which minimum l values of 0, 1/2 and 1 are found, but not the lmin=3/2 representations proposed by Biedenharn et al. (1972).

On a point source in an inhomogeneous medium

Abstract: Let , , be a second-order elliptic differential operator coinciding with the Laplace operator in a neighborhood of infinity. Let be the Green's function of the Cauchy problem for the operator . Under certain assumptions regarding the trajectories of the Hamiltonian system connected with the operator in question, the following results are obtained: 1) an asymptotic approximation with respect to smoothness to the function is constructed by Hadamard's method, 2) we show that the Fourier transformation of from to is an analytic function of in the complex plane with a cut along the negative part of the imaginary axis, and with and it gives the asymptotic behavior of the fundamental solution of the operator , 3) the asymptotic behavior as of the solutions of the nonstationary problem is obtained.Bibliography: 44 titles.

The problem of representing an arbitrary linear operator in the form of a differential operator of infinite order

Abstract: We consider linear operators, acting continuously from the space\(A_{R_1 }\) of functions analytic in the disk ¦ z ¦

On cyclic vectors of weighted shifts

Abstract: Sufficient conditions on a sequence are given in order that the linear manifold spanned by its right translates is dense in certain Hilbert spaces of sequences.