Spectral generating operators for non-stationary processes

1976 ◽  
Vol 8 (4) ◽  
pp. 831-846 ◽  
Author(s):  
D. Tj⊘stheim
Keyword(s):  
Spectral Representation ◽  
Stationary Process ◽  
Adjoint Operator ◽  
Stationary Processes ◽  
Wide Sense ◽  
Linear Transformations ◽  
Generating Operators ◽  
The Time Domain ◽  
Image Position ◽  
Stationary Random Processes

A new method for obtaining spectral-like representations for a large class of non-stationary random processes is formulated. For a wide sense stationary process X(t) in continuous-time the spectral representation is generated by a self-adjoint operator H such that X(t)= eiHt X(0). Extending certain recently established operator identities for wide sense stationary processes, it is shown that similar operators exist for classes of non-stationary processes. The representation generated by such an operator has the form and it shares some of the properties of the wide sense stationary spectral representation: it is dual in a precisely defined sense to the time domain representation of X(t). There exists a class L of linear transformations of X(t) such that for G ∈ L for some function g determined by G.

Spectral generating operators for non-stationary processes

1976 ◽  
Vol 8 (04) ◽  
pp. 831-846 ◽  
Author(s):  
D. Tj⊘stheim
Keyword(s):  
Spectral Representation ◽  
Stationary Process ◽  
Adjoint Operator ◽  
Stationary Processes ◽  
Wide Sense ◽  
Linear Transformations ◽  
Generating Operators ◽  
The Time Domain ◽  
Image Position ◽  
Stationary Random Processes

A new method for obtaining spectral-like representations for a large class of non-stationary random processes is formulated. For a wide sense stationary process X(t) in continuous-time the spectral representation is generated by a self-adjoint operator H such that X(t)= eiHt X(0). Extending certain recently established operator identities for wide sense stationary processes, it is shown that similar operators exist for classes of non-stationary processes. The representation generated by such an operator has the form and it shares some of the properties of the wide sense stationary spectral representation: it is dual in a precisely defined sense to the time domain representation of X(t). There exists a class L of linear transformations of X(t) such that for G ∈ L for some function g determined by G.

Non-Stationary Processes and Spectrum

1968 ◽  
Vol 20 ◽  
pp. 1203-1206 ◽  
Author(s):  
K. Nagabhushanam ◽  
C. S. K. Bhagavan
Keyword(s):  
Spectral Density ◽  
Density Function ◽  
Stationary Process ◽  
Spectral Density Function ◽  
Stationary Processes ◽  
Gaussian Stationary Process ◽  
Discrete Parameter ◽  
Image Position

In 1964, L. J. Herbst (3) introduced the generalized spectral density Function1for a non-stationary process {X(t)} denned by1where {η(t)} is a real Gaussian stationary process of discrete parameter and independent variates, the (a;)'s and (σj)'s being constants, the latter, which are ordered in time, having their moduli less than a positive number M.

On random processes that are almost strict sense stationary

1976 ◽  
Vol 8 (04) ◽  
pp. 820-830
Author(s):  
Dag Tj⊘stheim
Keyword(s):  
Stationary Process ◽  
Random Processes ◽  
Stationary Processes ◽  
Strict Sense ◽  
Wide Sense ◽  
Original Process ◽  
Extended Class ◽  
Gaussian Case

An extension of the class of strict sense stationary processes is studied. The extended class represents the strict sense analogy of an extension of wide sense stationary processes considered in an earlier paper [9]. The relations between the various types of processes defined are investigated in the general and in the Gaussian case, and some examples are given. It is shown that associated with a process belonging to the extended class there is a strict sense stationary process. The associated strict sense stationary process is unique iff the original process is ergodic.

scholarly journals Hilbert transforms and unitary equivalence

1967 ◽  
Vol 8 (2) ◽  
pp. 113-117
Author(s):  
R. Putnam
Keyword(s):  
Positive Measure ◽  
Spectral Representation ◽  
Closed Interval ◽  
Adjoint Operator ◽  
Unitary Equivalence ◽  
Hilbert Transforms ◽  
Cauchy Principal ◽  
Principal Value ◽  
Infinite Multiplicity ◽  
Image Position

If E is a subset of the real line of positive measure, then the associated Hilbert transform H = HE,where the integral is a Cauchy principal value, is a bounded self-adjoint operator on L2(E) (cf. Muskhelishvili [4]). In case E = (-∞, ∞) the transformation is also unitary with a spectrum consisting of 1 and -1, each of infinite multiplicity (Titchmarsh [10]). If E is a inite interval the spectral representation of H has been given by Koppelman and Pincus [3]; see also Putnam [6]. In particular the spectrum of H is in this case the closed interval [-1, 1]. Moreover, according to Widom [11], the spectrum of H is [-1, 1] whenever E ≠ (-∞, ∞), that is, whenever

On random processes that are almost strict sense stationary

1976 ◽  
Vol 8 (4) ◽  
pp. 820-830 ◽  
Author(s):  
Dag Tj⊘stheim
Keyword(s):  
Stationary Process ◽  
Random Processes ◽  
Stationary Processes ◽  
Strict Sense ◽  
Wide Sense ◽  
Original Process ◽  
Extended Class ◽  
Gaussian Case

An extension of the class of strict sense stationary processes is studied. The extended class represents the strict sense analogy of an extension of wide sense stationary processes considered in an earlier paper [9]. The relations between the various types of processes defined are investigated in the general and in the Gaussian case, and some examples are given. It is shown that associated with a process belonging to the extended class there is a strict sense stationary process. The associated strict sense stationary process is unique iff the original process is ergodic.

Analytic functions of finite dimensional linear transformations

1959 ◽  
Vol 55 (1) ◽  
pp. 51-61 ◽  
Author(s):  
S. N. Afriat
Keyword(s):  
Power Series ◽  
Integral Formula ◽  
Characteristic Root ◽  
Interpolation Formula ◽  
General Function ◽  
Linear Transformations ◽  
Multiple Characteristic ◽  
Characteristic Roots ◽  
Image Position ◽  
Necessary And Sufficient

Since the first introduction of the concept of a matrix, questions about functions of matrices have had the attention of many writers, starting with Cayley(i) in 1858, and Laguerre(2) in 1867. In 1883, Sylvester(3) defined a general function φ(a) of a matrix a with simple characteristic roots, by use of Lagrange's interpolation formula, and Buchheim (4), in 1886, extended his definition to the case of multiple characteristic roots. Then Weyr(5) showed in 1887 that, for a matrix a with characteristic roots lying inside the circle of convergence of a power series φ(ζ), the power series φ(a) is convergent; and in 1900 Poincaré (6) obtained the formulaefor the sum, where C is a circle lying in and concentric with the circle of convergence, and containing all the characteristic roots in its ulterior, such a formula having effectively been suggested by Frobenius(7) in 1896 for defining a general function of a matrix. Phillips (8), in 1919, discovered the analogue, for power series in matrices, of Taylor's theorem. In 1926 Hensel(9) completed the result of Weyr by showing that a necessary and sufficient condition for the convergence of φ(a) is the convergence of the derived series φ(r)(α) (0 ≼ r < mα; α) at each characteristic root α of a, of order r at most the multiplicity mα of α. In 1928 Giorgi(10) gave a definition, depending on the classical canonical decomposition of a matrix, which is equivalent to the contour integral formula, and Fantappie (11) developed the theory of this formula, and obtained the expressionfor the characteristic projectors.

Alternating projections and interpolation of stationary processes

1992 ◽  
Vol 29 (4) ◽  
pp. 921-931 ◽  
Author(s):  
Mohsen Pourahmadi
Keyword(s):  
Missing Values ◽  
Stationary Process ◽  
Dimensional Space ◽  
Stationary Processes ◽  
Finite Dimensional Space ◽  
Alternating Projections ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Explicit Formulae ◽  
Linear Interpolator

By using the alternating projection theorem of J. von Neumann, we obtain explicit formulae for the best linear interpolator and interpolation error of missing values of a stationary process. These are expressed in terms of multistep predictors and autoregressive parameters of the process. The key idea is to approximate the future by a finite-dimensional space.

A Purity Criterion for Pairs of Linear Transformations

1974 ◽  
Vol 26 (3) ◽  
pp. 734-745 ◽  
Author(s):  
Uri Fixman ◽  
Frank A. Zorzitto
Keyword(s):  
Perturbation Methods ◽  
Eigenvalue Problems ◽  
Vector Spaces ◽  
Infinite Dimension ◽  
Linear Transformations ◽  
Complex Vector ◽  
Algebraic Investigation ◽  
Image Position

In connection with the study of perturbation methods for differential eigenvalue problems, Aronszajn put forth a theory of systems (X, Y; A, B) consisting of a pair of linear transformations A, B:X → Y (see [1]; cf. also [2]). Here X and Y are complex vector spaces, possibly of infinite dimension. The algebraic aspects of this theory, where no restrictions of topological nature are imposed, where developed in [3] and [5]. We hasten to point out that the category of C2-systems (definition in § 1) in which this algebraic investigation takes place is equivalent to the category of all right modules over the ring of matrices of the form

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