Polynomial approximations in a generalized Nyman–Beurling criterion

We prove variants of Wiener's Tauberian theorem in the framework of quantum harmonic analysis, i.e. for convolutions between an absolutely integrable function and a trace class operator, or of two trace class operators. Our results include Wiener's Tauberian theorem as a special case. Applications of our Tauberian theorems are related to localization operators, Toeplitz operators, isomorphism theorems between Bargmann-Fock spaces and quantization schemes with consequences for Shubin's pseudodifferential operator calculus and Born-Jordan quantization. Based on the links between localization operators and Tauberian theorems we note that the analogue of Pitt's Tauberian theorem in our setting implies compactness results for Toeplitz operators in terms of the Berezin transform. In addition, we extend the results on Toeplitz operators to other reproducing kernel Hilbert spaces induced by the short-time Fourier transform, known as Gabor spaces. Finally, we establish the equivalence of Wiener's Tauberian theorem and the condition in the characterization of compactness of localization operators due to Fernández and Galbis.

We consider the problem of determining the Fourier integral in the Hilbert space of square integrable func- tions. Fourier integral is the scalar product of two functions belonging to the Hilbert space of square integrable functions and the Hilbert space of almost periodic functions. Scalar product for different Hilbert spaces defined at the intersection of these spaces, which contains only one zero element. Therefore, the Fourier integral is not defined in the Hilbert space of square integrable functions with nonzero norm.

Three energy principles for magnetostatic stability are known that are supposed to give necessary and sufficient conditions. For this reason their minimization has been the subject of a lot of work in plasma physics. Indeed one can easily justify the sufficiency of the conditions of stability devised from these energy principles. But to the demonstrate their necessity one usually assumes that the operators of the perturbed linearized motions have a complete spectrum of eigenfunctions in the space of square integrable functions. We show the weakness of that assumption and propose two new demonstrations of the necessity of energy principles for stability that require less stringent assumptions. The first demonstration involves some properties of the Laplace transform. In the second one we use the integral invariants of the linearized motion. The conclusions in both cases are identical: if one finds a trial function which makes the potential energy negative, the equilibrium is unstable. We give lower and upper bounds for the growth rate of the unstable perturbation.

The Kac-Baker model describes a 1-dim. classical lattice spin system with exponentially fast decaying two body interaction. The model was introduced by M. Kac and G. Baker to investigate the phenomenon of phase transition in systems with weak long-range interactions like van der Waals gas. Ruelle's dynamical zeta function for this model can be expressed in terms of Fredholm determinants of two transfer operators and hence is a meromorphic function. One of the two operators, found by M. Kac, is an integral operator with symmetric kernel acting in the Hilbert space of square integrable functions on the line. The other one is Ruelle's transfer operator acting in some Banach space of holomorphic observables of the system. In this paper we show how the Kac operator can be explicitly related basically through the Segal-Bargmann transform to the Ruelle operator restricted to a certain Segal-Bargmann space of entire functions in the complex plane. This allows us to show that Ruelle's zeta function for the Kac-Baker model has infinitely many non-trivial zeros on the real axis. In a special case we can show the existence of also infinitely many trivial zeros on the line Re s = ln 2 in the complex s-plane. Hence some kind of Riemann hypothesis seems to hold for this dynamical zeta function.

We construct a representation of the Weyl group in the p-adic Hilbert space of functions which are square integrable with respect to a p-adic valued Gaussian distribution. The operators corresponding to position and momentum are determined by groups of unitary operators with parameters restricted to some balls in the field of p-adic numbers. A surprising fact is that the radii of these balls are connected by `an uncertainty relation' which can be considered as a p-adic analogue of the Heisenberg uncertainty relations. The p-adic Hilbert space representation of the Weyl group is the basis for a calculus of pseudo-differential operators and for an operator quantization over p-adic numbers.

In present paper we prove an existence and give a moments estimate for the local time of Gaussian integrators. Every Gaussian integrator is associated with a continuous linear operator in the space of square integrable functions via white noise representation. Hence, all properties of such process are completely characterized by properties of the corresponding operator. We describe the sufficient conditions on continuous linear non-invertible operator which allow the local time of the integrator to exists at any real point. Moments estimate for local time is obtained. A continuous dependence of local time of Gaussian integrators on generating them operators is established. The received statement improves our result presented in [1].

Extensions of Wiener's Tauberian theorem for positive measures

Wiener's tauberian theorems for the Fourier–Bessel transformation and uniform approximation by RBF networks of Delsarte translates

Harmonic analysis on a locally compact group is the study of certain spaces and algebras of functions and measures defined on the group. The function spaces in question are defined by measurability and integrability conditions; manipulation of the relevant measures requires delicate techniques from measure and integration theory. Also, the fundamental theorems of the theory of representations of locally compact groups [see §22] depend wholly upon the study of a certain space of square integrable functions. Thus the theories of measure and integration play a vital role in the present book.

ABSTRACTWe study the stochastic homogenization and obtain a random fluctuation theory for semilinear elliptic equations with a rapidly varying random potential. To first order, the effective potential is the average potential and the nonlinearity is not affected by the randomness. We then study the limiting distribution of the properly scaled homogenization error (random fluctuations) in the space of square integrable functions, and prove that the limit is a Gaussian distribution characterized by homogenized solution, the Green’s function of the linearized equation around the homogenized solution, and by the integral of the correlation function of the random potential. These results enlarge the scope of the framework that we have developed for linear equations to the class of semilinear equations.

Chapter 7 On the Global Weak Solutions to a Variational Wave Equation

Diffusion multi-compartment models (MCM) allow for a fine and comprehensive study of the white matter microstructure. Non linear registration of MCM images may provide valuable information on the brain e.g. through population comparison. State-of-the-art MCM registration however relies on pairing-based similarity measures where the one-to-one mapping of MCM compartments is required. This approach leads to non differentiabilties or discontinuities, which may turn into poorer registration. Moreover, these measures are often specific to one MCM compartment model. We propose two new MCM similarity measures based on the space of square integrable functions, applied to MCM characteristic functions. These measures are pairing-free and agnostic to compartment types. We derive their analytic expressions for multi-tensor models and propose a spherical approximation for more complex models. Evaluation is performed on synthetic deformations and inter-subject registration, demonstrating the robustness of the proposed measures.

We construct a family of generalized coherent states attached to Landau levels of a charged particle moving in the Poincaré disc under a perpendicular uniform magnetic field. The corresponding coherent state transforms enable us to connect, by an integral transform, spaces of bound states of the particle with the space of square integrable functions on the real line. The established connection provides us with a new way to obtain hyperbolic Landau states.

The three-body Schrodinger operator in the space of square integrable functions is found to be a certain extension of operators which generate the exponential unitary group containing a subgroup with nilpotent Lie algebra of length \({\kappa + 1, \kappa = 0, 1, \ldots}\) As a result, the solutions to the three-body Schrodinger equation with decaying potentials are shown to exist in the commutator subalgebras. For the Coulomb three-body system, it turns out that the task is to solve—in these subalgebras—the radial Schrodinger equation in three dimensions with the inverse power potential of the form \({r^{-{\kappa}-1}}\) . As an application to Coulombic system, analytic solutions for some lower bound states are presented. Under conditions pertinent to the three-unit-charge system, obtained solutions, with \({\kappa = 0}\) , are reduced to the well-known eigenvalues of bound states at threshold.

Using normalized Hermite functions, we construct bases in the space of square integrable functions on the unit circle (L2(C)) and in l2(Z), which are related to each other by means of the Fourier transform and the discrete Fourier transform. These relations are unitary. The construction of orthonormal bases requires the use of the Gramm–Schmidt method. On both spaces, we have provided ladder operators with the same properties as the ladder operators for the one-dimensional quantum oscillator. These operators are linear combinations of some multiplication- and differentiation-like operators that, when applied to periodic functions, preserve periodicity. Finally, we have constructed riggings for both L2(C) and l2(Z), so that all the mentioned operators are continuous.