Curve In Hilbert Space Research Articles

ON A CLASS OF NONSTATIONARY CURVES IN HILBERT SPACE

Stationary random processes have been studied quite well over recent years starting with the works of A. N. Kolmogorov. The possibility of building nonstationary random process correlation theory was considered in the monographs by M. S. Livshits, A. A. Yantsevich, V. A. Zolotarev and others. Some classes of nonstationary curves were investigated by V. E. Katsnelson. In this paper nonstationary random processes are represented as curves in Hilbert space which "slightly deviate" from random processes with the correlation function of special kind. The infinitesimal correlation function has been introduced; in essence, this function characterizes the deviation from the correlation process with the given correlation function. The paper discusses the cases of nonstationary random processes, the operator of which has one‑dimensional imaginary component. Cases of a dissipative operator with descrete spectrum are also considered in this work. It is shown that the nonstationarity of the random process is closely related to the deviation of the operator from its conjugated operator. Using the triangle and universal models of non‑self‑ajoint operators it is possible to obtain the representation for the correlation function in the case of nonstationary process which replaces the Bochner – Khinchin representation for stationary random processes. The expresson for the infinitesimal correlation function was obtained for different cases of operator spectrum: for the descrete spectrum placed in the upper half‑plane and for the contrast‑free spectrum at zero. In the case of dissipative operator with descrete spectrum the infinitesimal function can be found in terms of special lambda function. For Lebesque spaces of compex‑valued squared integratable functions the expresson of infinitesimal function was found in terms of special zero order modified Bessel function. It was shown that a similar approach can be applied for the evolutionarily represented sequences in Hilbert spaces.

The Traveling Salesman Theorem for Jordan Curves in Hilbert Space

The Traveling Salesman Theorem for Jordan Curves in Hilbert Space

STATIONARY CONNECTED CURVES IN HILBERT SPACES

In this article the structure of non-stationary curves which are stationary connected in Hilbert space is studied using triangular models of non-self-adjoint operator. The concept of evolutionary representability plays here an important role. It is proved that if one of two curves in Hilbert space is evolutionary representable and the curves are stationary connected, then another curve is evolutionary representable too. These curves are studied firstly. The structure of a cross-correlation function in the case when operator, defining the evolutionary representation, has one-dimensional non-Hermitian subspace (the spectrum is discreet and situated in the upper complex half-plane or has infinite multiplicity at zero (Volterra operator)) is studied.

On Runge approximation in a Banach space

We show that on some open sets, more general than balls, Runge approximation is possible in certain Banach spaces, and also in certain complex Banach manifolds. We also show that there is an entire holomorphic curve in Hilbert space on which there is a bounded holomorphic function on the trace of a ball that has no bounded holomorphic extension to even a smaller concentric ball. Using the same technique we also prove that a form of Runge approximation better than an error function is not always possible.

The quantum geometric phase as a transformation invariant

The kinematic approach to the theory of the geometric phase is outlined. This phase is shown to be the simplest invariant under natural groups of transformations on curves in Hilbert space. The connection to the Bargmann invariant is brought out, and the case of group representations described.

Nonstationary curves in Hilbert spaces and their correlation functions II

One stohastic process after A. Kolmogorov can be considered as a curve in Hilbert space. The stationary random curves, i. e. such curves, the correlation functions (CF) of which depend on the difference of the arguments, have been studied by S. Bohner [1] and A. Hinchin [2]. These authors have obtained spectral representation of the stohastic stationary processes in the form

Self-Similar Probability Distributions

Self-Similar Probability Distributions

The orthogonal invariants of a curve in Hilbert space

Let x ( t ) , y ( t ) ( t ∈ R ) x(t),y(t)(t \in R) be a pair of continuous curves in H, not passing through the origin. We consider the problem of deciding when one curve is obtained by rotating the other one.

The Orthogonal Invariants of a Curve in Hilbert Space

The Orthogonal Invariants of a Curve in Hilbert Space

A characterization of oscillatory processes and their prediction

The oscillatory stochastic processes recently studied by Priestley are characterized as deformed stationary curves in a Hilbert space. This characterization leads to the simple time domain proof of prediction and moving average representation for these stochastic processes in terms of the associated stationary curve. In [6], A. N. Kolmogorov studied second order stationary processes as curves in Hilbert space. The idea again occurred in Cramer [2] for nonstationary processes. In this note we characterize the oscillatory processes introduced by Priestley [8] as deformed stationary curves in a Hilbert space and give a simple geometric solution for the prediction problem for such curves. As a consequence of this one can easily derive the analytic results on prediction in [1], thus providing a simple and more general solution for the prediction problem of oscillatory processes. Also our characterization makes the definition of the time dependent spectral distribution an obvious generalization of the stationary case. For the sake of being specific we consider throughout the continuous parameter case. 1. DEFINITION. Let H be a Hilbert-space and R be the space of real numbers with the usual topology. We say that (a) x is a continuous curve in H, if x is a continuous map of R into H. (b) y is a stationary continuous curve in H if (i) y is a continuous curve in H, (ii) (y(t), y(s)) is a function of t-s. With every curve x in H we associate the following subspaces of H: Hx(t) = S{x(r), 7 C t},2 Hx(+ oo) = {x(-), c RI, H$(-oo) = n H$(t). Received by the editors April 26, 1971. AMS 1970 subject classqiications. Primary 60G25; Secondary 47A50.

Multiplicity and Representation Theory of Purely Non-Deterministic Stochastic Processes

Multiplicity and Representation Theory of Purely Non-Deterministic Stochastic Processes

Stochastic Processes As Curves in Hilbert Space

Regular complex-valued random processes $x(t)$ with finite moments of second order are studied by methods of Hilbert space geometry. A representation formula (4) is given for the process $x(t)$ in terms of “past and present innovations”. The number N is called the complete spectral multiplicity of the process $x(t)$ and is the smallest number for which such a representation exists. It is shown that the multiplicity of $x(t)$ is uniquely determined by the corresponding correlation function and that one can always find a harmonizing process $x(t)$ which has the multiplicity prescribed in advance.