Search

Signals in real applications are typically finite in duration, dynamic and non-stationary processes with frequency characteristics varying over time. This often requires techniques capable of locally analyzing and processing signals. An integral transform known as the Stockwell transform is a combination of the classic Gabor transform and the current and versatile wavelet transform. It allows more accurate detection of subtle changes and easy interpretation in the time-frequency domain. In this paper, we study the mathematical underpinnings of the Stockwell transform. We look at the Stockwell transform as a stack of simple pseudo-differential operators parameterized by frequencies and give a complete description of the Stockwell spectra. [ABSTRACT FROM AUTHOR]

Motivated by problems in signal analysis, we define a class of time-frequency representations which is based on the short-time Fourier transform and depends on two fixed windows. We show that this class can be viewed as a link between the classical Rihaczek representation and the spectrogram. Correspondingly we formulate for this class a suitable general form of the uncertainty principle which have, as limit case, the uncertainty principles for the Rihaczek representation and for the spectrogram. We finally consider the questions of marginal distributions. We compute them in terms of convolutions with the windows and prove simple conditions for which average and standard deviation of the distributions in our class coincide with that of their marginals. [ABSTRACT FROM AUTHOR]

Using the boundedness of the maximal singular integral operator related to the Carleson-Hunt theorem we prove the boundedness and study the compactness of pseudo-differential operators a(x,D) with bounded measurable V (ℝR)-valued symbols a(x, ·) on the Lebesgue spaces Lp(ℝ) with 1 < p < δ, where V (ℝ) is the Banach algebra of all functions of bounded total variation on R. Replacement of absolutely continuous functions of bounded total variation by arbitrary functions of bounded total variation allows us to study pseudo-differential operators with symbols admitting discontinuities of the first kind with respect to the spatial and dual variables. Appearance of discontinuous symbols leads to non-commutative algebras of Fredholm symbols. Three different Banach algebras of pseudo-differential operators with discontinuous symbols acting on the spaces Lp(ℝ) are studied. We construct Fredholm symbol calculi for these algebras and establish Fredholm criteria for the operators in these algebras in terms of their Fredholm symbols. For the operators in the first algebra we also obtain an index formula. An application to the Haseman boundary value problem is given. [ABSTRACT FROM AUTHOR]

Let M(ω)p,q be the modulation space with parameters p, q and weight function ω. We prove that if t ∈ R, p, pj, q, qj ∈ [1, ∞], ω1, ω2 and ω are appropriate, and a ∈ M(ω)p,q, then the pseudo-differential operator at(x,D) is continuous from M(ω)p1,q1 to M(ω)p2,q2. If in addition pj = qj = 2, then we establish necessary and sufficient conditions on p and q in order to at(x,D) should be a Schatten-von Neumann operator of certain order. [ABSTRACT FROM AUTHOR]

In this paper a result of continuity for pseudo-differential operators with non-regular symbols on spaces of quasi-homogeneous type is given. More precisely, the symbols a(x, ξ) take their values in a quasi-homogeneous Besov space with respect to the x variable; moreover a finite number of derivatives with respect to the second variable satisfies, in Besov norm, decay estimates of quasi-homogeneous type. [ABSTRACT FROM AUTHOR]

For the differential operator P of order m and the inhomogeneous data f ∈ S′ on Rn, we say that the Lp-extension of the solution holds if u ∈ Lp, m ≤ n(1 − 1/p), and Pu = f on Rn / 0 imply Pu = f on Rn. In this article, we discuss which kind of inhomogeneous terms f ∈ S′ admit the Lp-extension of the solution. In previous works, this problem was studied by using classical Bochner's method ([1]) or a new method developed by the author and Uchida ([5], [4]). We consider inhomogeneous terms which are not covered by these results. [ABSTRACT FROM AUTHOR]

In this paper we study the local solvability in Gevrey classes for degenerate parabolic operators of order ≥ 2. We assume that the lower order term vanishes at a suitably smaller rate with respect to the principal part; we then analyze its influence on the behavior of the operator, proving local solvability in Gevrey spaces Gs for small s, and local nonsolvability in Gs for large s. [ABSTRACT FROM AUTHOR]

We study the exponential decay of solutions to differential equations of hypoelliptic type (see Definition 2.1). In particular we find sufficient conditions on the differential operator A in order for estimates of the kind $$ e^{\varepsilon \left\langle x \right\rangle ^r } Au \in V \Rightarrow e^{\varepsilon \left\langle x \right\rangle ^r } u \in V $$ to hold, for several types of functions and distribution spaces V. For example, V may be . [ABSTRACT FROM AUTHOR]

We construct the wave kernels of the non-isotropic twisted Laplacian by means of its heat kernel. We then express the wave kernels at the origin in terms of complex Fourier integrals and we exploit the connections of the phase functions with the complex integrals. [ABSTRACT FROM AUTHOR]

Basic properties of Fourier integral operators on the torus $$ \mathbb{T}^n = (\mathbb{R}/2\pi \mathbb{Z})^n $$ are studied by using the global representations by Fourier series instead of local representations. The results can be applied in studying hyperbolic partial differential equations. [ABSTRACT FROM AUTHOR]

The Plancherel formula and the inversion formula for Weyl transforms on compact and Hausdorff groups are given. A formula expressing the relationships of the wavelet constant, the degree of the irreducible and unitary representation and the volume of an arbitrary compact and Hausdorff group is derived. The role of the Weyl transforms in the derivation of the formulas for the heat kernels of Laplacians on compact Lie groups is explicated. The Green functions and the Riemann zeta functions of Laplacians on compact Lie groups are constructed using the corresponding heat kernels. [ABSTRACT FROM AUTHOR]

The method of construction of the fundamental solution for heat equations using pseudo-differential operators with parameter time variable is discussed, which is applicable to calculate traces of operators. This gives extensions of both the Gauss-Bonnet-Chern Theorem and the Riemann-Roch Theorem. [ABSTRACT FROM AUTHOR]

We present an overview of some of our recent results on the existence of rays of minimal growth for elliptic cone operators and two new results concerning the necessity of certain conditions for the existence of such rays. [ABSTRACT FROM AUTHOR]

We investigate operators on manifolds with edges from the point of view of the symbolic calculus induced by the singularities. We discuss new aspects of the quantization of edge-degenerate symbols which lead to continuous operators in weighted edge spaces. [ABSTRACT FROM AUTHOR]

For a class of pairs of entire matrix functions the null space of the natural analogue of the classical resultant matrix is described in terms of the common Jordan chains of the defining entire matrix functions. The main theorem is applied to two inverse problems. The first concerns convolution integral operators on a finite interval with matrix valued kernel functions and complements earlier results of [6]. The second is the inverse problem for matrix-valued continuous analogues of Szegő orthogonal polynomials. [ABSTRACT FROM AUTHOR]

We study the behavior of the sequence of minimal vectors corresponding to certain classes of operators on L2 spaces, including weighted composition operators such as those induced by Möbius transformations. In conjunction with criteria for quasinilpotence, the convergence of sequences associated with the minimal vectors leads to the construction of hyperinvariant subspaces. [ABSTRACT FROM AUTHOR]

The paper is concerned with a sequence of constants which appear in several problems. These problems include the minimal eigenvalue of certain positive definite Toeplitz matrices, the minimal eigenvalue of some higher-order ordinary differential operators, the norm of the Green kernels of these operators, the best constant in a Wirtinger-Sobolev inequality, and the conditioning of a special least squares problem. The main result of the paper gives the asymptotics of this sequence. [ABSTRACT FROM AUTHOR]

The central result of this paper is a method of characterizing those commuting tuples of operators that have a unitary dilation, in terms of the existence of a positive map with certain properties. Although this positivity condition is not necessarily easy to check given a concrete example, it can be used to find practical tests in some circumstances. As an application, we extend a dilation theorem of Sz.-Nagy and Foiaş concerning regular dilations to a more general setting [ABSTRACT FROM AUTHOR]

Notions of weak and uniformly weak mixing (to zero) are defined for bounded sequences in arbitrary Banach spaces. Uniformly weak mixing for vector sequences is characterized by mean ergodic convergence properties. This characterization turns out to be useful in the study of multiple recurrence, where mixing properties of vector sequences, which are not orbits of linear operators, are investigated. For bounded sequences, which satisfy a certain domination condition, it is shown that weak mixing to zero is equivalent with uniformly weak mixing to zero. [ABSTRACT FROM AUTHOR]

We describe a direct and transparent connection between the poly-Bergman type spaces on the upper half-plane and certain two-dimensional singular integral operators. [ABSTRACT FROM AUTHOR]

We present a survey of some results on the theory of singular integral operators with piece-wise continuous coefficients in the weighted spaces of continuous functions with a prescribed continuity modulus (generalized Hölder spaces Hω(Γ, ρ)) together with some new results related to oscillating (non-equilibrated) characteristics and oscillating weights. [ABSTRACT FROM AUTHOR]

Operator identities involving nonnegative selfadjoint operators play a fundamental role in interpolation theory and its applications. The theory is generalized here to selfadjoint operators whose negative spectra consist of a finite number of eigenvalues of finite total multiplicity. It is shown that such identities are closely associated with generalized Nevanlinna functions by means of the Kreĭn-Langer integral representation. The Potapov fundamental matrix inequality is generalized to this situation, and it is used to formulate and solve an operator interpolation problem analogous to the definite case. [ABSTRACT FROM AUTHOR]

The aim of the paper is to study the Fredholm property of pseudodifferential operators in the Sjöstrand class OPSw where we consider these operators as acting on the modulation spaces M2, p(ℝN). These spaces are introduced by means of a time-frequency partition of unity. The symbol class Sw does not involve any assumptions on the smoothness of its elements. In terms of their limit operators, we will derive necessary and sufficient conditions for operators in OPSw to be Fredholm. In particular, it will be shown that the Fredholm property and, thus, the essential spectra of operators in this class are independent of the modulation space parameter p ∈ (1, ∞). [ABSTRACT FROM AUTHOR]

An estimate for the dimension of the kernel of the singular integral operator with shift $$ \left( {I + \sum\limits_{j = 1}^n {a_j (t)U^j } } \right)P_ + + P_ - :L_2 (\mathbb{R}) \to L_2 (\mathbb{R}) $$ is obtained, where P± are the Cauchy projectors, (U ψ)(t) = ψ(t+h), h ∈ ℝ+, is the shift operator and aj(t) are continuous functions on the one point compactification of ℝ. The roots of the polynomial $$ 1 + \sum\limits_{j = 1}^n {a_j (\infty )\eta ^j } $$ are assumed to belong all simultaneously either to the interior of the unit circle or to its exterior. [ABSTRACT FROM AUTHOR]

Let (B(H) ‖·‖) be the algebra of all bounded operators on an infinite-dimensional Hilbert space H. Let B(H)sa be the set of all selfadjoint operators in B(H). Throughout the paper we denote by α a compact subset of ℝ and by B(H)sa(α) the set of all operators in B(H)sa with spectrum in α: $$ B(H)_{sa} (\alpha ) = \{ A = A^* \in B(H): Sp(A) \subseteq \alpha \} . $$ We will use similar notations Asa, Asa(α) for a Banach *-algebra A. Each bounded Borel function g on α defines, via the spectral theorem, a map A → g(A) from B(H)sa(α) into B(H). Various smoothness conditions when imposed on this map define the corresponding classes of operator-smooth functions. [ABSTRACT FROM AUTHOR]

Let V (ℝ) denote the Banach algebra of absolutely continuous functions of bounded total variation on ℝ. We study an algebra $$ \mathfrak{B} $$ of pseudodifferential operators of zero order with compound slowly oscillating V (ℝ)-valued symbols (x, y) ↦ a(x, y, ·) of limited smoothness with respect to x, y ∈ ℝ. Sufficient conditions for the boundedness and compactness of pseudodifferential operators with compound symbols on Lebesgue spaces Lp(ℝ) are obtained. A symbol calculus for the algebra $$ \mathfrak{B} $$ is constructed on the basis of an appropriate approximation of symbols by infinitely differentiable ones and by use of the techniques of oscillatory integrals. A Fredholm criterion and an index formula for pseudodifferential operators A ∈ $$ \mathfrak{B} $$ are obtained. These results are carried over to Mellin pseudodifferential operators with compound slowly oscillating V (ℝ)-valued symbols. Finally, we construct a Fredholm theory of generalized singular integral operators on weighted Lebesgue spaces Lp with slowly oscillating Muckenhoupt weights over slowly oscillating Carleson curves. [ABSTRACT FROM AUTHOR]

We find Fredholm criteria and a formula for the index of an arbitrary operator in the Banach algebra of singular integral operators with piecewise continuous coefficients on Nakano spaces (generalized Lebesgue spaces with variable exponent) with Khvedelidze weights over either Lyapunov curves or Radon curves without cusps. These results "localize" the Gohberg-Krupnik Fredholm theory with respect to the variable exponent. [ABSTRACT FROM AUTHOR]

We consider difference quotient operators in de Branges Hilbert spaces of entire functions. We give a description of the spectrum and a formula for the spectral subspaces. The question of completeness of the system of eigenvectors and generalized eigenvectors is discussed. For certain cases the s-numbers and the Schmidt-representation of the operator under discussion is explicitly determined. [ABSTRACT FROM AUTHOR]

Split Levinson and Schur algorithms for the inversion of centrosymmetric Toeplitz-plus-Hankel matrices are designed that work, in contrast to previous algorithms, for matrices with any rank profile. Furthermore, it is shown that the algorithms are related to generalized ZW-factorizations of the matrix and its inverse. [ABSTRACT FROM AUTHOR]

The inverse scattering problem is reviewed for determining the cross sectional area of a human vocal tract. Various data sets are examined resulting from a unit-amplitude, monochromatic, sinusoidal volume velocity sent from the glottis towards the lips. In case of nonuniqueness from a given data set, additional information is indicated for the unique recovery. [ABSTRACT FROM AUTHOR]

We study the set MΣ of all generalized positive self-adjoint solutions (that may be unbounded and have an unbounded inverse) of the KYP (Kalman-Yakubovich-Popov) inequality for a infinite-dimensional linear time-invariant system Σ in continuous time with scattering supply rate. It is shown that if MΣ is nonempty, then the transfer function of Σ coincides with a Schur class function in some right half-plane. For a minimal system Σ the converse is also true. In this case the set of all H ∈ MΣ with the property that the system is still minimal when the original norm in the state space is replaced by the norm induced by H is shown to have a minimal and a maximal solution, which correspond to the available storage and the required supply, respectively. The notions of strong H-stability, H-*-stability and H-bistability are introduced and discussed. We show by an example that the various versions of H-stability depend crucially on the particular choice of H ∈ MΣ. In this example, depending on the choice of the original realization, some or all H ∈ MΣ will be unbounded and/or have an unbounded inverse. [ABSTRACT FROM AUTHOR]

We consider arbitrary families of unbounded normal operators, commuting in a strong sense, in particular algebras consisting of unbounded normal operators, and investigate their connections with some algebras of fractions of continuous functions on compact spaces. New examples and properties of spaces of fractions are also given. [ABSTRACT FROM AUTHOR]

‘Hierarchical Semi-separable' matrices (HSS matrices) form an important class of structured matrices for which matrix transformation algorithms that are linear in the number of equations (and a function of other structural parameters) can be given. In particular, a system of linear equations Ax = b can be solved with linear complexity in the size of the matrix, the overall complexity being linearly dependent on the defining data. Also, LU and ULV factorization can be executed ‘efficiently', meaning with a complexity linear in the size of the matrix. This paper gives a survey of the main results, including a proof for the formulas for LU-factorization that were originally given in the thesis of Lyon [1], the derivation of an explicit algorithm for ULV factorization and related Moore-Penrose inversion, a complexity analysis and a short account of the connection between the HSS and the SSS (sequentially semi-separable) case. A direct consequence of the computational theory is that from a mathematical point of view the HSS structure is ‘closed' for a number operations. The HSS complexity of a Moore-Penrose inverse equals the HSS complexity of the original, for a sum and a product of operators the HSS complexity is no more than the sum of the individual complexities. [ABSTRACT FROM AUTHOR]

Canonical forms are given for pairs of quaternionic matrices, or equivalently matrix pencils, with various symmetry properties, under strict equivalence and symmetry respecting congruence. Symmetry properties are induced by involutory antiautomorphisms of the quaternions which are different from the quaternionic conjugation. Some applications are developed, in particular, canonical forms for quaternionic matrices that are symmetric or skewsymmetric with respect to symmetric or skewsymmetric quaternion-valued inner products. Another application concerns joint numerical cones of pairs of skewsymmetric quaternionic matrices. [ABSTRACT FROM AUTHOR]

In this paper we construct a class of homogeneous Hilbert modules over the disc algebra $$ \mathcal{A}(\mathbb{D}) $$ as quotients of certain natural modules over the function algebra $$ \mathcal{A}(\mathbb{D}^2 ) $$. These quotient modules are described using the jet construction for Hilbert modules. We show that the quotient modules obtained this way, belong to the class Bk($$ \mathbb{D} $$) and that they are mutually inequivalent, irreducible and homogeneous. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study the truncated matricial moment problem on a finite closed interval in the case of an odd number of prescribed moments by using of the FMI method of V.P. Potapov. The solvability of this problem is characterized by the fact that two block Hankel matrices built from the data of the problem are nonnegative Hermitian (Theorem 1.3). An essential step to solve the problem under consideration is to derive an effective coupling identity between both block Hankel matrices (Proposition 2.5). In the case that these Hankel matrices are both positive Hermitian we parametrize the set of solutions via a linear fractional transformation the generating matrix-valued function of which is a matrix polynomial whereas the set of parameters consists of distinguished pairs of meromorphic matrix-valued functions. [ABSTRACT FROM AUTHOR]

We review our recent work on the Schur transformation for scalar generalized Schur and Nevanlinna functions. The Schur transformation is defined for these classes of functions in several situations, and it is used to solve corresponding basic interpolation problems and problems of factorization of rational J-unitary matrix functions into elementary factors. A key role is played by the theory of reproducing kernel Pontryagin spaces and linear relations in these spaces. [ABSTRACT FROM AUTHOR]

In this chapter, we investigate asymptotic properties of one-parameter positive semigroups in L1(Ω, Σ, μ), where (Ω, Σ, μ) is a measure space with a σ-finite measure μ. In the last section, we shall also consider the theory of Markov semigroups in so-called non-commutative L1-spaces. For one-parameter positive semigroups in L1-spaces, there is a rich theory, which includes many results on the existence of invariant densities, criteria for asymptotic stability, decomposition theorems, etc. (cf. [71]). The choice of results presented in this chapter is motivated mainly by the author’s research interests, and it does not reflect the present state of the very broad asymptotic theory of positive semigroups in L1-spaces. We send the reader for many other important aspects of this theory and for their applications to books of Foguel [43], Krengel [67], Lasota and Mackey [71], and Schaefer [110]. [ABSTRACT FROM AUTHOR]

In this chapter, we deal with one-parameter positive semigroups in ordered Banach spaces. Firstly, we discuss the notion of ideally ordered Banach spaces and uniformly order convex Banach spaces. Both classes include Lp-spaces (1 ≤ p < ∞) as well as preduals of von Neumann algebras. We prove several theorems about positive semigroups in such Banach spaces. Then we consider positive semigroups in Banach lattices and investigate several types of asymptotic regularity of these semigroups. In the last section of this chapter, we deal with relations between the geometry of Banach lattices and mean ergodicity of bounded positive semigroups in them. [ABSTRACT FROM AUTHOR]

In the first chapter of the book, we give an introduction to the theory of one-parameter operator semigroups. We begin with the splitting theorem of Jacobs- Deleeuw-Glicksberg and the Eberlein mean ergodic theorem for a one-parameter operator semigroup. Then we present the elementary theory of C0-semigroups and discuss some relations between spectral properties of the generator of a C0- semigroup and its asymptotic behavior. We follow the standard textbooks [13], [48], [57], [67], [74], [80], [130], and send the reader for other deep and delicious topics of this theory to those books and to [17], [67], [87], [41], [89]. In the last section, we discuss the asymptotically finite-dimensional semigroups. We use frequently well-known results from operator theory and functional analysis, and send the reader to standard textbooks [2], [74], [105], and [130] for them. [ABSTRACT FROM AUTHOR]

Let T be an operator on a Banach space X. By an orbit of T we mean a sequence (Tnx)n∞=0, where x ∈ X is a fixed vector. This notion, which originated in the theory of dynamical systems, is closely related to the concepts of local spectral radius and capacity of an operator. Further motivations come from stability problems of semigroups of operators and the invariant subspace problem. [ABSTRACT FROM AUTHOR]

In this chapter we introduce and study another important spectral system for commuting operators — the Taylor spectrum. Although the definition of the Taylor spectrum is rather complicated, the Taylor spectrum has a distinguished property among other spectral systems, namely the existence of the functional calculus for functions analytic on a neighbourhood of the Taylor spectrum. From this reason many experts consider the Taylor spectrum to be the proper generalization of the ordinary spectrum for single operators. [ABSTRACT FROM AUTHOR]

In this chapter we study various types of essential spectra of operators on a Banach space X. They are closely connected with the Calkin algebra $$ \mathcal{B}\left( X \right)/\mathcal{K}\left( X \right) $$ , where $$ \mathcal{K}\left( X \right) $$ denotes the ideal of compact operators. [ABSTRACT FROM AUTHOR]

In this chapter we study the Banach algebra of all operators acting on a Banach space. All Banach spaces are assumed to be complex and non-trivial, of dimension at least 1. By an operator we always mean a bounded linear mapping between two Banach spaces. [ABSTRACT FROM AUTHOR]

In this chapter we study spectral theory in Banach algebras. Basic concepts and classical results are summarized in the first two sections. In the subsequent sections we study the approximate point spectrum, which is one of the most important examples of a spectrum in Banach algebras. The approximate point spectrum is closely related with the notions of removable and non-removable ideals. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Clifford algebra-valued admissible wavelets, which are associated to more than 2-dimensional Euclidean group with Dilations. We give an explicit characterization of the admissibility condition in terms of the Fourier transform, study the properties of this kind of wavelet transform, also give a family of admissible wavelets. [ABSTRACT FROM AUTHOR]

Weyl-Heisenberg frames are used to obtain matrix representations of wavelet multipliers. Simple test cases are used to illustrate the numerical computations of wavelet multipliers by means of the corresponding truncated matrices. [ABSTRACT FROM AUTHOR]

The concept of a stationary random field can be extended from the classical situation to a more general set up by letting the time parameter run through groups, homogeneous spaces or hypergroups. In the present exposition the author is concerned with spectral representations for stationary random fields over these algebraic-topological structures. [ABSTRACT FROM AUTHOR]

A unified account of a noncommutative operator trigonometry originated in 1966 by this author and its further developments and applications to date will be given within a format of a historical trace. Applications to wavelet and multiscale theories are included. A viewpoint toward possible future enlargement will be fostered. [ABSTRACT FROM AUTHOR]

We show how fundamental ideas from signal processing, multiscale theory and wavelets may be applied to nonlinear dynamics. The problems from dynamics include iterated function systems (IFS), dynamical systems based on substitution such as the discrete systems built on rational functions of one complex variable and the corresponding Julia sets, and state spaces of subshifts in symbolic dynamics. Our paper serves to motivate and survey our recent results in this general area. Hence we leave out some proofs, but instead add a number of intuitive ideas which we hope will make the subject more accessible to researchers in operator theory and systems theory. [ABSTRACT FROM AUTHOR]