Extremum Problem for the Wiener–Hopf Equation

The extremum problem for the Wiener–Hopf equation obtained by replacing the condition u(x) = 0, x < 0, by the condition of the minimum of the quadratic functional of the function u(x)exp(–x), – < x < , is solved in closed form.     

Paley–Wiener–Schwartz type theorem for the wavelet transform

In the present paper, we discuss about extension of the wavelet transform on distribution space of compact support and develop the Paley–Wiener–Schwartz type theorem for the wavelet transform on the same. Furthermore, Paley–Wiener–Schwartz type theorem for the wavelet transform is also established using the relation between the wavelet transform and double Fourier transform.     

Combined separable signals based neuro-fuzzy Hammerstein–Wiener model

Hammerstein–Wiener model can describe a large number of complicated industrial processes. In this paper, a novel identification method for neuro-fuzzy based Hammerstein–Wiener model is presented. A neuro-fuzzy system with correlation analysis based non-iterative parameter updating algorithm is proposed to model the static nonlinearity of Hammerstein–Winer processes. As a result, the proposed method not only avoid the inevitable restrictions on static nonlinear function encountered by using the polynomial approach, but also overcomes the problems of initialization and convergence of the model parameters, which are usually resorted to trial and error procedure in the existing iterative algorithms used for the identification of Hammerstein–Winer model. In addition, combined separable signals are adopted to identify the Hammerstein–Wiener process, resulting in the identification problem of the linear model separated from that of nonlinear parts. Moreover, one part of the input signals is extended to more general signals, such as binary signals, Gaussian signals or other modulated signals. Examples are used to illustrate the effectiveness of the proposed method.     

Paley–Wiener Theorems on the Siegel Upper Half-Space

Journal of Fourier Analysis and Applications - In this paper we study spaces of holomorphic functions on the Siegel upper half-space $${\mathcal U}$$ and prove Paley–Wiener type theorems for...     

A Wiener–Wintner theorem for the Hilbert transform

We prove the following extension of the Wiener–Wintner theorem and the Carleson theorem on pointwise convergence of Fourier series: For all measure-preserving flows (X,μ,T _t ) and f∈L ^p (X,μ), there is a set X _f ⊂X of probability one, so that for all x∈X _f ,

Wiener–Hopf equation whose kernel is a probability distribution

We prove the existence of a solution of an inhomogeneous generalized Wiener–Hopf equation whose kernel is a probability distribution on R generating a random walk drifting to +∞, while the inhomogeneous term f of the equation belongs to the space L ₁(0,∞) or L _∞(0,∞). We establish the asymptotic properties of the solution of this equation under various assumptions about the inhomogeneity f.     

Wiener–Itô Theorem in Terms of Wick Tensors

Wick tensors are used to describe homogeneous chaos and to define multiple Wiener integrals. The Wiener–Itô decomposition is expressed by the formula $$\varphi (x) = \sum\limits_{n = 0}^\infty {\frac{1}{{n!}}\int_B {[D^n (\mu \varphi )(0)]^ \sim}} (x + iy,...,x + iy)d\mu (y)$$ . We use this formula to interpret Hida's original idea of generalized multiple Wiener integrals as generalized functions acting on the space of test functions.     

The Paley–Wiener Theorem and Limits of Symmetric Spaces

We extend the Paley–Wiener theorem for Riemannian symmetric spaces to an important class of infinite-dimensional symmetric spaces. For this we define a notion of propagation of symmetric spaces and examine the direct (injective) limit symmetric spaces defined by propagation. This relies on some of our earlier work on invariant differential operators and the action of Weyl group invariant polynomials under restriction.     

The Paley–Wiener theorem for certain nilpotent Lie groupsJean

We generalize the classical Paley–Wiener theorem to special types of connected, simply connected, nilpotent Lie groups: First we consider nilpotent Lie groups whose Lie algebra admits an ideal which is a polarization for a dense subset of generic linear forms on the Lie algebra. Then we consider nilpotent Lie groups such that the co-adjoint orbits of all the elements of a dense subset of the dual of the Lie algebra 𝔤^* are flat (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Paley–Wiener theorem for the Weinstein transform and applications

In this paper our aim is to establish the Paley–Wiener Theorem for the Weinstein Transform. Furthermore, some applications are presents. In particular some properties for the generalized translation operator associated with the Weinstein operator are proved, an integral representation and a series representation for a function in the Paley–Wiener classes are investigated.     

Multivariate polynomial interpolation and sampling in Paley–Wiener spaces

In this paper, an equivalence between existence of particular exponential Riesz bases for spaces of multivariate bandlimited functions and existence of certain polynomial interpolants for functions in these spaces is given. Namely, polynomials are constructed which, in the limiting case, interpolate ${\left\{({\tau }_n,f\left({\tau }_n\right))\right\}}_n$ for certain classes of unequally spaced data nodes ${\left\{{\tau }_n\right\}}_n$ and corresponding ${?}_2$ sampled data ${\left\{f\left({\tau }_n\right)\right\}}_n$. Existence of these polynomials allows one to construct a simple sequence of approximants for an arbitrary multivariate bandlimited function $f$ which demonstrates $L_2$ and uniform convergence on ${\mathbb{R}}^d$ to $f$. A simpler computational version of this recovery formula is also given at the cost of replacing $L_2$ and uniform convergence on ${\mathbb{R}}^d$ with $L_2$ and uniform convergence on increasingly large subsets of ${\mathbb{R}}^d$. As a special case, the polynomial interpolants of given ${?}_2$ data converge in the same fashion to the multivariate bandlimited interpolant of that same data. Concrete examples of pertinent Riesz bases and unequally spaced data nodes are also given.     

Local Wiener–Hopf factorization and indices over arbitrary fields

In this paper, a generalization to arbitrary fields of the usual Wiener–Hopf equivalence of complex valued rational matrix functions is given and the left local Wiener–Hopf factorization indices defined in our previous work [A. Amparan, S. Marcaida, I. Zaballa, Local realizations and local polynomial matrix representations of systems, Linear Algebra Appl. 425 (2007) 757–775] are proved to form a complete system of invariants for this equivalence relation. For the case when the field is algebraically closed a reduced form of a controllable matrix pair under the feedback equivalence is presented for which the controllability indices can be written as sums of the local controllability indices [A. Amparan, S. Marcaida, I. Zaballa, On the existence of linear systems with prescribed invariants for system similarity, Linear Algebra Appl. 413 (2006) 510–533].     

Inclusions of Waterman–Shiba spaces into generalized Wiener classes

The characterization of the inclusion of Waterman–Shiba spaces ΛBV^(p)$\Lambda {\mathit{BV}}^{(p)}$ into generalized Wiener classes of functions BV(q;δ)$\mathit{BV}(q;\delta )$ is given. It uses a new and shorter proof and extends an earlier result of U. Goginava.     

A Note on Wiener–Hopf Factorization for Markov Additive Processes

We prove the Wiener–Hopf factorization for Markov additive processes. We derive also Spitzer–Rogozin theorem for this class of processes which serves for obtaining Kendall’s formula and Fristedt representation of the cumulant matrix of the ladder epoch process. Finally, we also obtain the so-called ballot theorem.     

Representation theory of finite groups in Wiener–Hopf factorization problem

We consider the Wiener–Hopf factorization problem for a matrix function that is completely defined by its first column: the succeeding columns are obtained from the first one by means of a finite group of permutations. The symmetry of this matrix function allows us to reduce the dimension of the problem. In particular, we find some relations between its partial indices and can compute some of the indices. In special cases, we can explicitly obtain the Wiener–Hopf factorization of the matrix function.     

Minimality and (weighted) interpolation in Paley–Wiener spaces

We investigate links between minimality, Carleson condition, and (weighted) interpolation in Paley–Wiener spaces. In particular, we show that the Carleson condition on a sequence Λ together with minimality in Paley–Wiener spaces ${PW_{\tau}^{p}}$ implies the interpolation property of Λ in ${PW_{\tau+\epsilon}^{p}}$ , for every ${\epsilon > 0}$ . This result does not, surprisingly, require uniform minimality.     

Revisiting Landauʼs density theorems for Paley–Wiener spaces

We present a surprisingly simple approach to Landau?s density theorems for sampling and interpolation, which provides stronger versions of these results. In particular, we extend the interpolation theorem to unbounded spectra.