Uncertainty principles for the weinstein transform

The Weinstein transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization and a variant of Cowling-Price theorem, Miyachi’s theorem, Beurling’s theorem, and Donoho-Stark’s uncertainty principle are obtained for the Weinstein transform.     

An analogue of Beurling’s theorem for the Jacobi transform

In this paper, we prove Beurling＇s theorem for the Jacobi transform, from which we derive some other versions of uncertainty principles.     

Beurling’s theorem for SL(2,$${\mathbb{R}}$$)

We prove Beurling’s theorem for the full group SL(2,). This is the master theorem in the quantitative uncertainty principle as all the other theorems of this genre follow from it.     

Dirichlet forms for singular one-dimensional operators and on graphs

We treat the time evolution of states on a finite directed graph, with singular diffusion on the edges of the graph and glueing conditions at the vertices. The operator driving the evolution is obtained by the method of quadratic forms on a suitable Hilbert space. Using the Beurling–Deny criteria we describe glueing conditions leading to positive and to submarkovian semigroups, respectively.     

VITALI-HAHN-SAKS THEOREM FOR VECTOR MEASURES ON OPERATOR ALGEBRAS

We show that the direct generalization of the Vitali–Hahn–Sakstheorem is not valid for all measures on von Neumann algebras.By applying a general equicontinuity argument, we prove a directextension of the Vitali–Hahn–Saks theorem for awide range of vector measures on von Neumann algebra s and JBWalgebras. We also characterize relatively compact sets of vectormeasures on operator algebras.     

Statistical Properties of Endomorphisms and Compact Group Extensions

The statistical properties of endomorphisms under the assumptionthat the associated Perron–Frobenius operator is quasicompactare considered. In particular, the central limit theorem, weakinvariance principle and law of the iterated logarithm for sufficientlyregular observations are examined. The approach clarifies therole of the usual assumptions of ergodicity, weak mixing, andexactness. Sufficient conditions are given for quasicompactness of thePerron–Frobenius operator to lift to the correspondingequivariant operator on a compact group extension of the base.This leads to statistical limit theorems for equivariant observationson compact group extensions. Examples considered include compact group extensions of piecewiseuniformly expanding maps (for example Lasota–Yorke maps),and subshifts of finite type, as well as systems that are nonuniformlyexpanding or nonuniformly hyperbolic.     

On spectral analysis in locally compact motion groups

Beurling's theorem in spectral analysis of bounded functions on the real line is generalized to a class of locally compact motion groups and to Heisenberg groups.     

Weyl’s theorem and thin spectra

Using the notion of thin sets we prove a theorem of Weyl type for the Wolf essential spectrum ofT ∈β (H). *Further we show that Weyl’s theorem holds for a restriction convexoid operator and consequently modify some results of Berberian. Finally we show that Weyl’s theorem holds for a paranormal operator and that a polynomially compact paranormal operator is a compact perturbation of a diagnoal normal operator. A structure theorem for polynomially compact paranormal operators is also given.     

Uniform Laws of Large Numbers for Triangular Arrays of Function-indexed Processes under Random Entropy Conditions

Mean Glivenko Cantelli Theorems are established for triangular arrays of rowwise independent processes. Methods developed by Pollard (1990) are combined with a truncation method essentially due to Alexander (1987). By this, applicability to partial sum processes in particular is achieved, for which Pollard’s truncation method fails. Nevertheless, the metric entropy condition appearing here is kept as weak as Pollard’s by means of application of Hoffmann-Jørgensen’s inequality, which has not been used so far in this context. The main theorem of the paper contains Pollard’s theorem as well as former results by Giné and Zinn (1984) and proves applicable to so-called random measure processes, certain function-indexed processes including empirical processes, partial-sum processes, the sequential empirical process and certain types of smoothed empirical processes. Statistical applications include nonparametric regression and the estimation of the intensity measure of a spatial Poisson process (Poisson point process).     

Borel complexity and computability of the Hahn–Banach Theorem

The classical Hahn–Banach Theorem states that any linear bounded functional defined on a linear subspace of a normed space admits a norm-preserving linear bounded extension to the whole space. The constructive and computational content of this theorem has been studied by Bishop, Bridges, Metakides, Nerode, Shore, Kalantari Downey, Ishihara and others and it is known that the theorem does not admit a general computable version. We prove a new computable version of this theorem without unrolling the classical proof of the theorem itself. More precisely, we study computability properties of the uniform extension operator which maps each functional and subspace to the set of corresponding extensions. It turns out that this operator is upper semi-computable in a well-defined sense. By applying a computable version of the Banach–Alaoglu Theorem we can show that computing a Hahn–Banach extension cannot be harder than finding a zero in a compact metric space. This allows us to conclude that the Hahn–Banach extension operator is -computable while it is easy to see that it is not lower semi-computable in general. Moreover, we can derive computable versions of the Hahn–Banach Theorem for those functionals and subspaces which admit unique extensions. This work has been partially supported by the National Research Foundation (NRF) Grant FA2005033000027 on “Computable Analysis and Quantum Computing”. An extended abstract version has been published in the conference proceedings [7].     

A free boundary problem for the Laplacian with a constant Bernoulli-type boundary condition

We study a free boundary problem for the Laplace operator, where we impose a Bernoulli-type boundary condition. We show that there exists a solution to this problem. We use A. Beurling’s technique, by defining two classes of sub- and super-solutions and a Perron argument. We try to generalize here a previous work of A. Henrot and H. Shahgholian. We extend these results in different directions.     

Optimal control for cooperative parabolic systems governed by Schrodinger operator with control constraints

^** Email: Bahaa_gm{at}hotmail.com A distributed control problem for cooperative parabolic systemsgoverned by Schrödinger operator is considered. The performanceindex is more general than the quadratic one and has an integralform. Constraints on controls are imposed. Making use of theDubovitskii–Milyutin theorem given by Walczak (1984, Onesome control problems. Acta Univ. Lod. Folia Math., 1, 187–196),the optimality conditions are derived for the Dirichlet problem.     

A complete analogue of Hardy’s theorem on SL2(ℝ) and characterization of the heat kernel

A theorem of Hardy characterizes the Gauss kernel (heat kernel of the Laplacian) on ℝ from estimates on the function and its Fourier transform. In this article we establisha full group version of the theorem for SL₂(ℝ) which can accommodate functions with arbitraryK-types. We also consider the ‘heat equation’ of the Casimir operator, which plays the role of the Laplacian for the group. We show that despite the structural difference of the Casimir with the Laplacian on ℝⁿ or the Laplace—Beltrami operator on the Riemannian symmetric spaces, it is possible to have a heat kernel. This heat kernel for the full group can also be characterized by Hardy-like estimates.     

On the Equivalence and Generalized of Weyl Theorem Weyl Theorem

We know that an operator T acting on a Banach space satisfying generalized Weyl＇s theorem also satisfies Weyl＇s theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl＇s theorem, then it also satisfies generalized Weyl＇s theorem. We give also a sinlilar result for the equivalence of a-Weyl＇s theorem and generalized a-Weyl＇s theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.     

Some remarks on linear functional differential inequalities of hyperbolic type

We prove that, for the validity of a certain theorem on differential inequalities for a linear functional differential equation of hyperbolic type {fx327-01} with a negative linear operator {fx327-02}, it is necessary that ℓ be an (a, c)-Volterra operator. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 283–292, February, 2008.     

Generalized projection operators in Geometric Algebra

Given an automorphism and an anti-automorphism of a semigroup of a Geometric Algebra, then for each element of the semigroup a (generalized) projection operator exists that is defined on the entire Geometric Algebra. A single fundamental theorem holds for all (generalized) projection operators. This theorem makes previous projection operator formulas [2] equivalent to each other. The class of generalized projection operators includes the familiar subspace projection operation by implementing the automorphism ‘grade involution’ and the anti-automorphism ‘inverse’ on the semigroup of invertible versors. This class of projection operators is studied in some detail as the natural generalization of the subspace projection operators. Other generalized projection operators include projections ontoany invertible element, or a weighted projection ontoany element. This last projection operator even implies a possible projection operator for the zero element.     

Weyl’s Theorem and Perturbations

In the present paper we examine the stability of Weyl’s theorem under perturbations. We show that if T is an isoloid operator on a Banach space, that satisfies Weyl’s theorem, and F is a bounded operator that commutes with T and for which there exists a positive integer n such that Fⁿ is finite rank, then T + F obeys Weyl’s theorem. Further, we establish that if T is finite-isoloid, then Weyl’s theorem is transmitted from T to T + R, for every Riesz operator R commuting with T. Also, we consider an important class of operators that satisfy Weyl’s theorem, and we give a more general perturbation results for this class.     

On the Norm of the Fourier—Gegenbauer Projection in Weighted L p Spaces

We extend the results of Pollard [4] and give asymptotic estimates for the norm of the Fourier—Gegenbauer projection operator in the appropriate weighted L _p space. In particular, we settle the question of whether the projection is bounded for p=(2λ+1)/λ and p=(2λ+1)/(λ+1) , where λ is the index for the family of Gegenbauer polynomials under consideration. March 19, 1997. Date revised: June 3, 1998. Date accepted: August 1, 1998.     

An Answer to S. Simons’ Question on the Maximal Monotonicity of the Sum of a Maximal Monotone Linear Operator and a Normal Cone Operator

The question whether or not the sum of two maximal monotone operators is maximal monotone under Rockafellar’s constraint qualification—that is, whether or not “the sum theorem” is true—is the most famous open problem in Monotone Operator Theory. In his 2008 monograph “From Hahn-Banach to Monotonicity”, Stephen Simons asked whether or not the sum theorem holds for the special case of a maximal monotone linear operator and a normal cone operator of a closed convex set provided that the interior of the set makes a nonempty intersection with the domain of the linear operator. In this note, we provide an affirmative answer to Simons’ question. In fact, we show that the sum theorem is true for a maximal monotone linear relation and a normal cone operator. The proof relies on Rockafellar’s formula for the Fenchel conjugate of the sum as well as some results featuring the Fitzpatrick function.