Uncertainty principles and optimally sparse wavelet transforms

In this paper we introduce a new localization framework for wavelet transforms, such as the 1D wavelet transform and the Shearlet transform. Our goal is to design nonadaptive window functions that promote sparsity in some sense. For that, we introduce a framework for analyzing localization aspects of window functions. Our localization theory diverges from the conventional theory in two ways. First, we distinguish between the group generators, and the operators that measure localization (called observables). Second, we define the uncertainty of a signal transform as a whole, instead of defining the uncertainty of an individual window. We show that the uncertainty of a window function, in the signal space, is closely related to the localization of the reproducing kernel of the wavelet transform, in phase space. As a result, we show that using uncertainty minimizing window functions, results in representations which are optimally sparse in some sense.     

Heisenberg evolution of quantum observables represented by unbounded operators

This paper deals with open quantum systems. In particular, we focus on the adjoint quantum master equations with initial conditions given by unbounded operators. Examples of this type of initial data are the position and momentum operators of quantum oscillators and the occupation number operator in many-body particle systems. The article establishes the existence and uniqueness of solutions of the operator equations governing the motion of unbounded observables under the Born-Markov approximations. To this end, we develop the relation between operator evolution equations arising in quantum mechanics and stochastic evolutions equations of Schrödinger type. Furthermore, we examine quantum dynamical semigroup properties of the Heisenberg evolutions of general classes of observables.     

Extensions,dilations, and spectral problems of singular Hamiltonian systems

In this paper, we construct a space of boundary values for minimal symmetric 1D Hamiltonian operator with defect index (1,1) (in limit‐point case at a(b) and limit‐circle case at b(a)) acting in the Hilbert space In terms of boundary conditions at a and b, all maximal dissipative, accumulative, and self‐adjoint extensions of the symmetric operator are given. Two classes of dissipative operators are studied. They are called “dissipative at a” and “dissipative at b.” For 2 cases, a self‐adjoint dilation of dissipative operator and its incoming and outgoing spectral representations are constructed. These constructions allow us to establish the scattering matrix of dilation and a functional model of the dissipative operator. Further, we define the characteristic function of the dissipative operators in terms of the Weyl‐Titchmarsh function of the corresponding self‐adjoint operator. Finally, we prove theorems on completeness of the system of root vectors of the dissipative operators.     

Limiting Absorption Principle for Stratified Operators with Thresholds: A General Method

Out problem is about propagation of waves in stratified strips. The operators are quite general, a typical example being a coupled elasto-acoustic operator H defined in ?² × I where I is a bounded interval of ? with coefficients depending only on z ∈ I. The “conjugate operator method” will be applied to an operator obtained by a spectral decomposition of the partial Fourier transform ? of H. Around each value of the spectrum (except the eigenvalues) including the thresholds, a conjugate operator is constructed which permits to get the ”good properties“ of regularity for H. A limiting absorption principle is then obtained for a large class of operators at every point of the spectrum (except eigenvalues).     

Difference operators,Green's matrix,sampling theory and applications in signal processing

This paper introduces sampling representations for discrete signals arising from self adjoint difference operators with mixed boundary conditions. The theory of linear operators on finite-dimensional inner product spaces is employed to study the second-order difference operators. We give necessary and sufficient conditions that make the operators self adjoint. The equivalence between the difference operator and a Hermitian Green's matrix is established. Sampling theorems are derived for discrete transforms associated with the difference operator. The results are exhibited via illustrative examples, involving sampling representations for the discrete Hartley transform. Families of discrete fractional Fourier-type transforms are introduced with an application to image encryption.     

Full Fourier-Bessel transform and the algebra of singular pseudodifferential operators

The one-dimensional full Fourier-Bessel transform was introduced by I.A. Kipriyanov and V.V. Katrakhov on the basis of even and odd small (normalized) Bessel functions. We introduce a mixed full Fourier-Bessel transform and prove an inversion formula for it. Singular pseudodifferential operators are introduced on the basis of the mixed full Fourier-Bessel transform. This class of operators includes linear differential operators in which the singular Bessel operator and its (integer) powers or the derivative (only of the first order) of powers of the Bessel operator act in one of the directions. We suggest a method for constructing the asymptotic expansion of a product of such operators. We present the form of the adjoint singular pseudodifferential operator and show that the constructed algebra is, in a sense, a *-algebra.     

A sharper uncertainty principle

The work strengthens the result established by L. Cohen on uncertainty principle involving phase derivative. We propose stronger uncertainty principles not only in the classical setting for Fourier transform, but also for self-adjoint operators. We also deduce the conditions that give rise to the equal relation of the uncertainty principle. Examples are provided to show that the new uncertainty principle is truly sharper than the existing ones in literature.     

Localized Frames and Compactness

We introduce the concept of weak localization for continuous frames and use this concept to define a class of weakly localized operators. This class contains many important classes of operators, including: short time Fourier transform multipliers, Calderon–Toeplitz operators, Toeplitz operators on various functions spaces, Anti-Wick operators, some pseudodifferential operators, some Calderon–Zygmund operators, and many others. In this paper, we study the boundedness and compactness of weakly localized operators. In particular, we provide a characterization of compactness for weakly localized operators in terms of the behavior of their Berezin transforms.     

Unique solvability of pseudohyperbolic equations with singular right-hand sides

In this paper, we consider the solvability of the Cauchy problem for pseudohyperbolic equations (partial differential equations of third order). For the case in which the right-hand side is a generalized function (distribution) of finite order, we establish a theorem on the unique solvability for a sufficiently general pseudohyperbolic operator. The method of proof is based on a specially constructed “scale” of a priori inequalities for the direct and adjoint operators.     

Quantum Fields as Pointlike Localized Objects

We show that Wightman fields are always well defined objects at each space-time point in the meaning of sesquilinear forms. These sesquilinear forms “sandwiched” with e^–cH (H is the energy operator) are closable operators. We use these results to extend some assertions of Fredenhagen and Hertel about the recovering of Wightman fields from a Haag -Kastler theory of local observables.     

On characterizing the set of possible effective tensors of composites: The variational method and the translation method

A general algebraic framework is developed for characterizing the set of possible effective tensors of composites. A transformation to the polarization-problem simplifies the derivation of the Hashin-Shtrikman variational principles and simplifies the calculation of the effective tensors of laminate materials. A general connection is established between two methods for bounding effective tensors of composites. The first method is based on the variational principles of Hashin and Shtrikman. The second method, due to Tartar, Murat, Lurie, and Cherkaev, uses translation operators or, equivalently, quadratic quasiconvex functions. A correspondence is established between these translation operators and bounding operators on the relevant non-local projection operator, T₁. An important class of bounds, namely trace bounds on the effective tensors of two-component media, are given a geometrical interpretation: after a suitable fractional linear transformation of the tensor space each bound corresponds to a tangent plane to the set of possible tensors. A wide class of translation operators that generate these bounds is found. The extremal translation operators in this class incorporate projections onto spaces of antisymmetric tensors. These extremal translations generate attainable trace bounds even when the tensors of the two-components are not well ordered. In particular, they generate the bounds of Walpole on the effective bulk modulus. The variational principles of Gibiansky and Cherkaev for bounding complex effective tensors are reviewed and used to derive some rigorous bounds that generalize the bounds conjectured by Golden and Papanicolaou. An isomorphism is shown to underlie their variational principles. This isomorphism is used to obtain Dirichlet-type variational principles and bounds for the effective tensors of general non-selfadjoint problems. It is anticipated that these variational principles, which stem from the work of Gibiansky and Cherkaev, will have applications in many fields of science.     

On some properties of a class of degenerate pseudodifferential operators

A new class of pseudodifferential operators with degeneration is considered. The operators are constructed using a special integral transform mapping a weighted differentiation operator to a multiplication operator. The composition and boundedness properties of such operators in special weighted spaces are examined. Theorems on commutation of such operators with differentiation operators are obtained. The behavior of these operators as t → 0and t → +∞ is investigated. The properties of adjoint operators are studied, and an analogue of Gårding’s inequality is proved.     

Optimization of Mayer Problem with Sturm–Liouville-Type Differential Inclusions

The present paper studies a new class of problems of optimal control theory with Sturm–Liouville-type differential inclusions involving second-order linear self-adjoint differential operators. Our main goal is to derive the optimality conditions of Mayer problem for differential inclusions with initial point constraints. By using the discretization method guaranteeing transition to continuous problem, the discrete and discrete-approximation inclusions are investigated. Necessary and sufficient conditions, containing both the Euler–Lagrange and Hamiltonian-type inclusions and “transversality” conditions are derived. The idea for obtaining optimality conditions of Mayer problem is based on applying locally adjoint mappings. This approach provides several important equivalence results concerning locally adjoint mappings to Sturm–Liouville-type set-valued mappings. The result strengthens and generalizes to the problem with a second-order non-self-adjoint differential operator; a suitable choice of coefficients then transforms this operator to the desired Sturm–Liouville-type problem. In particular, if a positive-valued, scalar function specific to Sturm–Liouville differential inclusions is identically equal to one, we have immediately the optimality conditions for the second-order discrete and differential inclusions. Furthermore, practical applications of these results are demonstrated by optimization of some “linear” optimal control problems for which the Weierstrass–Pontryagin maximum condition is obtained.     

Support size restrictions on time-frequency representations of functions on finite abelian groups

We obtain uncertainty principles for finite abelian groups that relate the cardinality of the support of a function to the cardinality of the support of its short–time Fourier transform. These uncertainty principles are based on well–established uncertainty principles for the Fourier transform. In terms of applications, the uncertainty principle for the short–time Fourier transform implies the existence of a class of equal norm tight Gabor frames that are maximally robust to erasures. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Compact adjoint operators and approximation properties

This paper is concerned with the space of all compact adjoint operators from dual spaces of Banach spaces into dual spaces of Banach spaces and approximation properties. For some topology on the space of all bounded linear operators from separable dual spaces of Banach spaces into dual spaces of Banach spaces, it is shown that if a bounded linear operator is approximated by a net of compact adjoint operators, then the operator can be approximated by a sequence of compact adjoint operators whose operator norms are less than or equal to the operator norm of the operator. Also we obtain applications of the theory and, in particular, apply the theory to approximation properties.     

Bogoliubov’s metric as a global characteristic of the family of metrics in the Hilbert algebra of observables

We comparatively analyze a one-parameter family of bilinear complex functionals with the sense of “deformed” Wigner-Yanase-Dyson scalar products on the Hilbert algebra of operators of physical observables. We establish that these functionals and the corresponding metrics depend on the deformation parameter and the extremal properties of the Kubo-Martin-Schwinger and Wigner-Yanase metrics in quantum statistical mechanics. We show that the Bogoliubov-Kubo-Mori metric is a global (integral) characteristic of this family. It occupies an intermediate position between the extremal metrics and has the clear physical sense of the generalized isothermal susceptibility. We consider the example for the SU(2) algebra of observables in the simplest model of an ideal quantum spin paramagnet.     

Trace Theorems for Anisotropic Weighted SOBOLEV Spaces in a Corner

In this paper, we prove a trace theorem for anisotropic weighted Sobolev spaces in a cube Q naturally associated to a class of degenerate elliptic operators. The fundamental property of this class is the existence of a suitable metric d which is “natural” for the operators. The basic tool of the proof is a representation formula obtained via suitable non-euclidean translations closely fitting the geometry of the d-balls. In a more particular situation, we construct a right inverse of the trace operator and we describe the compatibility conditions on the edges of Q.     

THE UNIQUENESS OF OPERATIONAL QUANTITIES IN VON NEUMANN ALGEBRAS

Abstract

Since 1970 a number of operational quantities, characteristic of either the semi-Fredholm operators or of some “ideal” of compact-like operators, have been introduced in the theory of bounded operators between Banach spaces and applied successfully to for example perturbation theory. More recently such quantities have been introduced even in the abstract setting of Fredholm theory in a von Neumann algebra relative to some closed two-sided ideal. We show that in this fairly general setting there is only one “reasonable” set of such quantities—a result which in its present form is to the best of our knowledge new even in the case of B(H), the algebra of all bounded operators on a Hilbert space H. We accomplish this by first of all introducing the concept of a (reduced) minimum modulus in the setting of C*-algebras and developing the relevant techniques. In the process we generalise a result of Nikaido [N].     

An alternative point of view to the theory of fractional Fourier transform

The concept of the fractional Fourier transform is framed withinthe context of quantum evolution operators. This point of viewyields an extension of the above concept and greatly simplifiesthe underlying operational algebra. It is also proved that amultidimensional extension can be performed by using a biorthogonalmultiindex harmonic oscillator basis. It is finally shown thatmost of the proposed physical interpretations of the fractionalFourier transform are just trivial consequences of the analysisdeveloped in this paper.     

Generalized fourier transform and its applications

We consider the generalized Fourier transform treated as an operator on the dual of an arbitrary locally convex space. We give a definition of this operator and establish its basic properties. Special attention is paid to cases in which the range of the generalized Fourier transform coincides with a weighted space of entire functions. The results are applied to finding the orders and types of operators in various spaces.