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We consider the Volterra integration operator $V$, $Vf(x) =\int_{0}^{x} f(t)\mkern1mu dt$, on the following subspace of the Wiener algebra $W[ 0,1]$: \[ W^{(1) }[ 0,1] :=\{ f\in W[ 0,1] : f^{\prime}\!\in W[ 0,1] \}. \] We investigate solvability of the operator equations $VA=\lambda AV$ and $VA=\lambda A^{2}V$, where $\lambda\in\mathbb{C}$ is a complex number. Our proof is based on the Duhamel product of functions defined by \[ (f \circledast g) (x) :=\frac{d}{dx}\int_{0}^{x}f(x-t) g(t) \, dt. \] [ABSTRACT FROM AUTHOR]

In this paper, we study the approximation characteristics of weighted p-Wiener algebra A ω p T d for 1 ≤ p < ∞ defined on the d-dimensional torus T d . In particular, we investigate the asymptotic behavior of the approximation numbers, Kolmogorov numbers, and entropy numbers associated with the embeddings i d : A ω p T d → A T d and i d : A ω p T d → L q T d for 1 ≤ p , q < ∞ , where A T d is the Wiener algebra defined on the d-dimensional torus T d . [ABSTRACT FROM AUTHOR]

We consider a Nevanlinna–Pick interpolation problem on finite sequences of the unit disc, constrained by Beurling–Sobolev norms. We find sharp asymptotics of the corresponding interpolation quantities, thereby improving the known estimates. On our way we obtain a S. M. Nikolskii type inequality for rational functions whose poles lie outside of the unit disc. It shows that the embedding of the Hardy space H2 into the Wiener algebra of absolutely convergent Fourier/Taylor series is invertible on the subset of rational functions of a given degree, whose poles remain at a given distance from the unit circle.

We use the q-Duhamel product to provide a Banach algebra structure to some closed subspaces of the Wiener disk- algebra W + D of analytic functions on the unit disk D of the complex plane C. We study the q-integration operator on W + D , namely, we characterize invariant subspaces of this operator and describe its extended eigenvalues and extended eigenvectors. Moreover, we prove an addition formula for the spectral multiplicity of the direct sum of q-integration operator on W + D and some Banach space operator. [ABSTRACT FROM AUTHOR]

In this paper we present parallel theories on constructing Wiener algebras in the bicomplex setting. With the appropriate symmetry condition, the bicomplex matrix valued case can be seen as a complex valued case and, in this matrix valued case, we make the necessary connection between bicomplex analysis and complex analysis with symmetry. We also write an application to superoscillations in this case. [ABSTRACT FROM AUTHOR]

We consider a Nevanlinna–Pick interpolation problem on finite sequences of the unit disc, constrained by Beurling–Sobolev norms. We find sharp asymptotics of the corresponding interpolation quantities, thereby improving the known estimates. On our way we obtain a S. M. Nikolskii type inequality for rational functions whose poles lie outside of the unit disc. It shows that the embedding of the Hardy space H2 into the Wiener algebra of absolutely convergent Fourier/Taylor series is invertible on the subset of rational functions of a given degree, whose poles remain at a given distance from the unit circle. [ABSTRACT FROM AUTHOR]

A representative class of matrix functions from the Wiener algebra of order 2 admitting an effective factorization is found. The factorization problem for elements of this class is reduced to a truncated Wiener–Hopf equation with a contracting integral operator. The latter guarantees, as will be shown in the article, the existence of a canonical factorization and its explicit construction for matrix functions from the class under consideration. [ABSTRACT FROM AUTHOR]

We generalize the results for Banach algebras of pseudodifferential operators obtained by Gröchenig and Rzeszotnik (Ann Inst Fourier 58:2279–2314, 2008) to quasi-algebras of Fourier integral operators. Namely, we introduce quasi-Banach algebras of symbol classes for Fourier integral operators that we call generalized metaplectic operators, including pseudodifferential operators. This terminology stems from the pioneering work on Wiener algebras of Fourier integral operators (Cordero et al. in J Math Pures Appl 99:219–233, 2013), which we generalize to our framework. This theory finds applications in the study of evolution equations such as the Cauchy problem for the Schrödinger equation with bounded perturbations, cf. (Cordero, Giacchi and Rodino in Wigner analysis of operators. Part II: Schrödinger equations, ). [ABSTRACT FROM AUTHOR]

In this paper, we study the approximation characteristics of weighted p-Wiener algebra AωpTd for 1≤p<∞ defined on the d-dimensional torus Td. In particular, we investigate the asymptotic behavior of the approximation numbers, Kolmogorov numbers, and entropy numbers associated with the embeddings id:AωpTd→ATd and id:AωpTd→LqTd for 1≤p, q<∞, where ATd is the Wiener algebra defined on the d-dimensional torus Td.

The classes of band-dominated operators and the subclass of operators in the Wiener algebra W are known to be inverse closed. This paper studies and extends partially known results of that type for one-sided and generalized invertibility. Furthermore, for the operators in the Wiener algebra W invertibility, the Fredholm property and the Fredholm index are known to be independent of the underlying space l p , 1 ≤ p ≤ ∞ . Here this is completed by the observation that even the kernel and a suitable direct complement of the range as well as generalized inverses of operators in W are invariant w.r.t. p. [ABSTRACT FROM AUTHOR]

A method for reducing the factorization problem for an arbitrary matrix function with nonnegative total index in (an everywhere dense subalgebra of) the Wiener algebra of order 2 to the truncated Wiener–Hopf equation is found. With the help of the method, an efficient factorization of one class of matrix functions in the Wiener algebra of order 2 is constructed. [ABSTRACT FROM AUTHOR]

We derive dispersion estimates for solutions of a one‐dimensional discrete Dirac equations with a potential. In particular, we improve our previous result, weakening the conditions on the potential. To this end we also provide new results concerning scattering for the corresponding perturbed Dirac operators which are of independent interest. Most notably, we show that the reflection and transmission coefficients belong to the Wiener algebra. [ABSTRACT FROM AUTHOR]

New relations between the Banach algebras of absolutely convergent Fourier integrals of complex-valued measures of Wiener and various issues of trigonometric Fourier series (see classical monographs by A. Zygmund [1] and N. K. Bary [2]) are described. Those bilateral interrelations allow one to derive new properties of the Fourier series from the known properties of the Wiener algebras, as well as new results to be obtained for those algebras from the known properties of Fourier series. For example, criteria, i.e. simultaneously necessary and sufficient conditions, are obtained for any trigonometric series to be a Fourier series, or the Fourier series of a function of bounded variation, and so forth. Approximation properties of various linear summability methods of Fourier series (comparison, approximation of function classes and single functions) and summability almost everywhere (often with the set indication) are considered. The presented material was reported by the author on 12.02.2021 at the Zoom-seminar on the theory of real variable functions at the Moscow State University. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new method for studying the inhomogeneous vector Riemann–Hilbert boundary value problem (which is also called the Riemann boundary value problem) in the Wiener algebra of order two. The method consists in reducing the Riemann problem to a truncated Wiener–Hopf equation (to a convolution equation on a finite interval). The idea of the method was proposed by the author in a previous work. Here the method is applied to the inhomogeneous Riemann boundary value problem and to matrix functions of a more general form. The efficiency of the method is shown in the paper: new sufficient conditions for the existence of a canonical factorization of the matrix function in the Wiener algebra of order two are obtained. In addition, it was established that for the correct solvability of the inhomogeneous vector Riemann boundary value problem, it is necessary and sufficient to prove the uniqueness of the solution to the corresponding truncated homogeneous Wiener–Hopf equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the homogeneous vector Riemann boundary value problem (the factorization problem) from a new point of view. Namely, we reduce the Riemann problem to the truncated Wiener–Hopf equation (a convolution equation in a finite interval). We establish a connection between the problem of the factorization of a matrix function in the Wiener algebra of order two and the truncated Wiener–Hopf equation and obtain an explicit formula for this relationship. Note that the form of the matrix function considered in this paper differs from its most general form in the Wiener algebra; however, in this case, this is inessential. The truncated Wiener–Hopf equation is one of the most thoroughly studied Fredholm integral equations of the second kind. Therefore, the idea of the mentioned reduction can be expected to lead to new results in studying the factorization problem. [ABSTRACT FROM AUTHOR]

The possible instability of partial indices is one of the important constraints in the creation of approximate methods for the factorization of matrix functions. This paper is devoted to a study of a specific class of triangular matrix functions given on the unit circle with a stable and unstable set of partial indices. Exact conditions are derived that guarantee a preservation of the unstable set of partial indices during a perturbation of a matrix within the class. Thus, even in this probably simplest of cases, when the factorization technique is well developed, the structure of the parametric space (guiding the types of matrix perturbations) is non-trivial. [ABSTRACT FROM AUTHOR]

There is little analytical theory for the behavior of solutions of the Kuramoto–Sivashinsky equation in two spatial dimensions over long times. We study the case in which the spatial domain is a two-dimensional torus. In this case, the linearized behavior depends on the size of the torus—in particular, for different sizes of the domain, there are different numbers of linearly growing modes. We prove that small solutions exist for all time if there are no linearly growing modes, proving also in this case that the radius of analyticity of solutions grows linearly in time. In the general case (i.e., in the presence of a finite number of growing modes), we make estimates for how the radius of analyticity of solutions changes in time. [ABSTRACT FROM AUTHOR]

Abstract We work with spaces of periodic functions on the d -dimensional torus. We show that estimates for L ∞ -approximation of Sobolev functions remain valid when we replace L ∞ by the isotropic periodic Besov space B ∞ , 1 0 or the periodic Besov space with dominating mixed smoothness S ∞ , 1 0 B. For t > 1 / 2 , we also prove estimates for L 2 -approximation of functions in the Besov space of dominating mixed smoothness S 1 , ∞ t B , describing exactly the dependence of the involved constants on the dimension d and the smoothness t. [ABSTRACT FROM AUTHOR]

A quasi-Toeplitz (QT) matrix is a semi-infinite matrix of the kind A = T (a) + E where T (a) = (a j − i) i , j ∈ ℤ + , E = (e i , j) i , j ∈ ℤ + is compact and the norms ∥ a ∥ W = ∑ i ∈ ℤ | a i | and ∥ E ∥ 2 are finite. These properties allow to approximate any QT matrix, within any given precision, by means of a finite number of parameters. QT matrices, equipped with the norm ∥ A ∥ Q T = α ∥ a ∥ W + ∥ E ∥ 2 , for α = (1 + 5) / 2 , are a Banach algebra with the standard arithmetic operations. We provide an algorithmic description of these operations on the finite parametrization of QT matrices, and we develop a MATLAB toolbox implementing them in a transparent way. The toolbox is then extended to perform arithmetic operations on matrices of finite size that have a Toeplitz plus low-rank structure. This enables the development of algorithms for Toeplitz and quasi-Toeplitz matrices whose cost does not necessarily increase with the dimension of the problem. Some examples of applications to computing matrix functions and to solving matrix equations are presented, and confirm the effectiveness of the approach. [ABSTRACT FROM AUTHOR]

In this paper we prove the global existence, uniqueness, optimal large time decay rates, and uniform gain of analyticity for the exponential PDE ht = Δe-Δh in the whole space Rdx. We assume the initial data is of medium size in the Wiener algebra A(Rd); we use the initial condition Δh0 ∊ A(Rd) which is scale-invariant with respect to the invariant scaling of the exponential PDE. This exponential PDE was derived in [18] and more recently in [22]. [ABSTRACT FROM AUTHOR]

Salem’s necessary conditions for a trigonometric series to be the Fourier series of an integrable function are generalized to the nonperiodic case for functions in the Wiener algebra. Applications of the obtained result are given. [ABSTRACT FROM AUTHOR]

In this paper a twofold inverse problem for orthogonal matrix functions in the Wiener class is considered. The scalar-valued version of this problem was solved by Ellis and Gohberg in 1992. Under reasonable conditions, the problem is reduced to an invertibility condition on an operator that is defined using the Hankel and Toeplitz operators associated to the Wiener class functions that comprise the data set of the inverse problem. It is also shown that in this case the solution is unique. Special attention is given to the case that the Hankel operator of the solution is a strict contraction and the case where the functions are matrix polynomials. [ABSTRACT FROM AUTHOR]

We present multiplicative factorizations of positive elements in decomposing ordered Banach algebras. In order to apply strong factorization results to the Wiener algebra, we introduce the concept of semi-strongly decomposing algebras. Several applications are considered; including factorizations of M-matrices, regular operators and Laurent polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we concern over a class of the Wigner–Poisson- X α system introduced and developed by Bao et al. (2003), Mauser (2001) and Stimming (2005), respectively. The model describes the quantum mechanical motion of particles under the influence of the nonlocal Hartree potential and a local power term (exchange potential). The existence and uniqueness of local mild solution to the n -dimensional ( n = 1 , 3 ) Cauchy problems are established on the space of some integrable functions whose inverse Fourier transforms are also integrable. [ABSTRACT FROM AUTHOR]

Using tools from the theory of operator ideals and s -numbers, we develop a general approach to transfer estimates for L 2 -approximation of Sobolev functions into estimates for L ∞ -approximation, with precise control of all involved constants. As an illustration, we derive some results for periodic isotropic Sobolev spaces H s ( T d ) and Sobolev spaces of dominating mixed smoothness H mix s ( T d ) , always equipped with natural norms. Some results for L p -approximation ( 2 < p < ∞ ) are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and uniqueness of the local mild solution to the Cauchy problem of the n-dimensional ( n≥3) Wigner-Poisson-BGK equation in the space of some integrable functions whose inverse Fourier transform are integrable. The main difficulties in establishing mild solution are to derive the boundedness and locally Lipschitz properties of the appropriate nonlinear terms in the Wiener algebra. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We introduce algebras which are inductive limits of Banach spaces and carry inequalities which are counterparts of the inequality for the norm in a Banach algebra, and show that the algebra of holomorphic functions on a compact set is such an algebra. We then define an associated Wiener algebra, and prove the corresponding version of the well-known Wiener theorem. Finally, we consider factorization theory in these algebras, and in particular, in the associated Wiener algebra. [ABSTRACT FROM AUTHOR]

In this paper we study band truncation approximations for operators in uniform Roe algebras of countable discrete groups. Under conditions on certain growth rates for discrete groups, we find large classes of dense subspaces of uniform Roe algebras whose elements can be approximated by their band truncations in the operator norm. We apply these results to construct a nested family of spectral invariant Banach algebras on discrete groups. For a group with polynomial growth, the intersection of these Banach algebras is a spectral invariant dense subalgebra of the uniform Roe algebra. For a group with subexponential growth, we show that the Wiener algebra of the group is a spectral invariant dense subalgebra of the uniform Roe algebra. [ABSTRACT FROM AUTHOR]

In this paper, we study various generalizations of the classical Wiener algebra on a Banach space and prove analogs of Wiener's theorem on the invertibility of elements of such algebras. [ABSTRACT FROM AUTHOR]

We prove that if the window function g is in the Wiener amalgam space or in the Lipschitz amalgam space then its canonical dual window $${\gamma = S^{-1}_{g} g}$$ is in the same space, where S is the Gabor frame operator. [ABSTRACT FROM AUTHOR]