Discrete Wiener-Hopf Equation Research Articles

On a discrete Wiener-Hopf equation

On a discrete Wiener-Hopf equation

The Discrete Wiener-Hopf Equation Whose Kernel is a Probability Distribution with Positive Drift

We consider the discrete Wiener-Hopf equation with inhomogeneous term $$g = \{ {g_j}\} _{j = 0}^\infty \in {l_\infty }$$; the kernel of the equation is an arithmetic probability distribution generating a random walk drifting to +∞. We prove that the previously obtained formula for the Wiener-Hopf equation with general arithmetic kernel for g ∈ l1 is a solution to the equation for g ∈ l∞ and that successive approximations converge to the solution. The asymptotics of the solution is established in the following cases with account taken of their peculiarities: (1) g ∈ l1; (2) g ∈ l∞; (3) gj → const as j → ∞; (4) g ∉ l1 and gj ↓ 0 as j → ∞.

The discrete Wiener–Hopf equation with submultiplicative asymptotics of the solution

The discrete Wiener–Hopf equation with submultiplicative asymptotics of the solution

The Discrete Wiener-Hopf Equation with Probability Kernel of Oscillating Type

We prove the existence of a solution to the discrete inhomogeneous Wiener-Hopf equation whose kernel is an arithmetic probability distribution generating an oscillating random walk. Asymptotic properties of the solution are established depending on the properties of the inhomogeneous term of the equation and its kernel.

The discrete Wiener–Hopf equation with probability kernel of oscillating type

The discrete Wiener–Hopf equation with probability kernel of oscillating type

Edge diffraction on triangular and hexagonal lattices: Existence, uniqueness, and finite section

The diffraction of lattice waves by two distinct types of defects, namely a rigid constraint and a crack, in the triangular and hexagonal (honeycomb) lattice models is analyzed. A semi-analytical solution of the discrete Helmholtz equation is provided everywhere in the lattice using the lattice Green’s function, when the length of the defect is finite, in all four problems. For the case of a semi-infinite defect, each problem involves the inversion of a Toeplitz operator on ℓp (1≤p≤∞), a truncation of which appears in the finite defect problem. It is shown that the symbol of the Toeplitz operator satisfies the Krein conditions and is continuous on the unit circle in the complex plane. The existence and uniqueness of solution of the discrete Wiener–Hopf equation in ℓp, corresponding to each of the four problems, follow and the method of finite section is applicable. It is established that the solution of each of the four discrete Sommerfeld problems is unique in ℓ2 and for the case of finite defect the displacement field, near any one tip, is approximated by its semi-infinite counterpart. The paper includes several graphical illustrations supplementary to the mathematical analysis presented and the main results established.

Projective method for the equation of risk theory in the arithmetic case

We consider a discrete model of operation of an insurance company whose initial capital can take any integer value. In this statement, the problem of nonruin probability is naturally solved by the Wiener–Hopf method. Passing to generating functions and reducing the fundamental equation of the risk theory to a Riemann boundary-value problem on the unit circle, we show that this equation is a special one-sided discrete Wiener–Hopf equation whose symbol has a unique zero and, furthermore, this zero is simple. On the basis of the constructed solvability theory for this equation, we justify the applicability of the projective method to the approximation of ruin probabilities in the spaces l+ 1 and c+ 0. Conditions for the distributions of waiting times and claims under which the method converges are established. The delayed renewal process and stationary renewal process are considered and approximations for the ruin probabilities in these processes are obtained.

Approximation methods for a class of discrete Wiener-Hopf equations

In this paper, we consider approximation methods for operator equations of the form \[Au + Bu = f,\] where \(A\) is a discrete Wiener-Hopf operator on \(l_p\) (\(1 \leq p \lt \infty\)) which symbol has roots on the unit circle with arbitrary multiplicities (not necessary integers). Conditions on perturbation \(B\) and \(f\) are given in order to guarantee the applicability of projection-iterative methods. Effective error estimates, and simultaneously, decaying properties for solutions are obtained in terms of some smooth spaces.

Operational Quadrature Methods for Wiener-Hopf Integral Equations

We study the numerical solution of Wiener-Hopf integral equations by a class of quadrature methods which lead to discrete Wiener-Hopf equations, with quadrature weights constructed from the Fourier transform of the kernel (or rather, from the Laplace transforms of the kernel halves). As the analytical theory of Wiener-Hopf equations is likewise based on the Fourier transform of the kernel, this approach enables us to obtain results on solvability and stability and error estimates for the discretization. The discrete Wiener-Hopf equations are solved by using an approximate Wiener-Hopf factorization obtained with FFT. Numerical experiments with the Milne equation of radiative transfer are included.

Operational quadrature methods for Wiener-Hopf integral equations

We study the numerical solution of Wiener-Hopf integral equations by a class of quadrature methods which lead to discrete Wiener-Hopf equations, with quadrature weights constructed from the Fourier transform of the kernel (or rather, from the Laplace transforms of the kernel halves). As the analytical theory of Wiener-Hopf equations is likewise based on the Fourier transform of the kernel, this approach enables us to obtain results on solvability and stability and error estimates for the discretization. The discrete Wiener-Hopf equations are solved by using an approximate Wiener-Hopf factorization obtained with FFT. Numerical experiments with the Milne equation of radiative transfer are included.

Spectral factorization of rectangular rational matrix functions with application to discrete Wiener-Hopf equations

The properties of a discrete Wiener-Hopf equation are closely related to the factorization of the symbol of the equation. We give a necessary and sufficient condition for existence of a canonical Wiener-Hopf factorization of a possibly nonregular rational matrix function W relative to a contour which is a positively oriented boundary of a region in the finite complex plane. The condition involves decomposition of the state space in a minimal realization of W and, if it is satisfied, we give explicit formulas for the factors. The results are generalized by means of centered realizations to arbitrary rational matrix functions. The proposed approach can be used to solve discrete Wiener-Hopf equations whose symbols are rational matrix functions which admit canonical factorization relative to the unit circle.

Solution space of discrete Wiener-Hopf equations and Grassmann manifold

It is shown that the solution space of a system of discrete Wiener-Hopf equations is a set of points on an infinite-dimensional Grassmann manifold. Fractional transformations acting on the solution space are also discussed.

On a fast algorithm for the approximate solution of the discrete Wiener-Hopf equation and an estimation of the accuracy

A fast algorithm for the approximate solution of the discrete Wiener-Hopf equation is described and convergence rate estimates are obtained. The numerical algorithm is constructed by using the discrete Fourier transformation.

A discussion of various approaches to the linear system identification problem

This paper deals with the pole zero identification of a linear system from a measured input-output record. One objective is to show that the pencil-of-function method minimizes a weighted version of the Kalman equation error. It follows that the pencil-of-function method is capable of yielding robust estimates for poles located in a given region of the complex s plane. The second objective of this paper is to illustrate that identical sets of equations arise in three supposedly different analytical techniques for obtaining the impulse response of a system. The techniques investigated are 1) the least squares technique based on the discrete Wiener-Hopf equation, 2) Pisarenko's eigenvalue method, and 3) Jain's pencil-of-function method. The proof of equivalence is valid only for the noise-free case when the system order is known. Instead of using the conventional differential equation formulation, equivalence is shown with the integral form utilized in the pencil-of-function method.

Über das Reduktionsverfahren für diskrete Wiener-Hopf-Gleichungen mit unstetigem Symbol

The results of the paper complete and improve some hitherto existing results about the reduction method for discrete Wiener-Hopf equations with discontinuous symbol and, on the other hand, illustrate some new aspects of this set of problems. Thus, it is proved that if the reduction method in $l^p(1 \\leq p < \\infty)$ is applicable to the Toeplitz operators $T(a_r), a_r \\in L^{\\infty} (r = 1, \\dots, R)$ and if $a_1, \\dots, a_R$ have no common singularities (discontinuities), then it is applicable to $T(a_1 \\dots a_R)$. Furthermore, there is proved that for a large class of piecewise continuous symbols the reduction method in $L^p$ is applicable to $T(a)$ if and only if both $T(a)$ and $T(\\bar {a})$ are invertible.

Realizable Wiener filtering in two dimensions

The extension of Wiener's classical mean-square estimation theory to a two-dimensional setting is presented. In analogy with the one-dimensional problem, the optimal realizable filter is derived by solution of a two-dimensional, discrete Wiener-Hopf equation using a spectral factorization procedure. Filters are developed for the cases of prediction, filtering, and smoothing, and appropriate error expressions are derived to characterize their performance.

Projection method for solving degenerate discrete Wiener-Hopf equations

Projection method for solving degenerate discrete Wiener-Hopf equations

The discrete Wiener-Hopf equation

The discrete Wiener-Hopf equation

The factorization of discrete-process spectral matrices

A technique is presented for salving the discrete version of the multidimensional Wiener-Hopf equation by spectral factorization. This equation is derived to establish a need for spectral factorization and to determine the requirements of the factors of the spectral matrix. The method of factoring the spectral matrix of continuous systems, developed by Davis, is then extended to discrete systems. More specifically, a matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H(z)</tex> must be found such that the matrix of the spectra of the input signals equals the product of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H(z^{-1})</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H^{T}(z)</tex> . A technique for finding this matrix is presented. The nonanticipatoriness as well as the stability of the elements of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H(z)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H^{-1}(z)</tex> must be and is guaranteed. It is then shown that the solution to the discrete Wiener-Hopf equation is unique.