Singular Value Decomposition of a Finite Hilbert Transform Defined on Several Intervals and the Interior Problem of Tomography: The Riemann‐Hilbert Problem Approach

We study the asymptotics of singular values and singular functions of a finite Hilbert transform (FHT), which is defined on several intervals. Transforms of this kind arise in the study of the interior problem of tomography. We suggest a novel approach based on the technique of the matrix Riemann‐Hilbert problem (RHP) and the steepest‐descent method of Deift‐Zhou. We obtain a family of matrix RHPs depending on the spectral parameter λ and show that the singular values of the FHT coincide with the values of λ for which the RHP is not solvable. Expressing the leading‐order solution as λ → 0 of the RHP in terms of the Riemann Theta functions, we prove that the asymptotics of the singular values can be obtained by studying the intersections of the locus of zeroes of a certain Theta function with a straight line. This line can be calculated explicitly, and it depends on the geometry of the intervals that define the FHT. The leading‐order asymptotics of the singular functions and singular values are explicitly expressed in terms of the Riemann Theta functions and of the period matrix of the corresponding normalized differentials, respectively. We also obtain the error estimates for our asymptotic results. © 2016 Wiley Periodicals, Inc.     

The Finite Hilbert Transform in ℒ2

It is well known that the finite HILBERT transform T is a NOETHER (FREDHOLM) operator when considered as a map from ?^p into itself if 1 < p < 2 or 2 < p < ∞. When p = 2, the map T is not a NOETHER operator. We present two theorems which characterize the range of T in ?² and, as immediate consequences, give simple expressions for its inverse.     

Two new formulas for the numerical evaluation of the Hilbert Transform

We develop two algorithms for the numerical evaluation of the semi-infinite Hilbert Transform of functions with a given algebraic behaviour at the origin and at infinity. The first algorithm is connected with a Gauss-Jacobi type quadrature formula for unbounded intervals; the second is based on a rational Bernstein-type operator. Error estimates for different classes of functions are shown. Finally numerical examples are given, comparing the rules among themselves.     

Bilinear Hilbert Transform on Measure Spaces

In this article we obtain the boundedness of the periodic, discrete and ergodic bilinear Hilbert transform, from , where 1$, and $p_3\ge 1$" align="middle" border="0"> . The main techniques are a bilinear version of the transference method of Coifman and Weiss and certain discretization of bilinear operators. In the periodic case, we also obtain the boundedness for      

Finite Intervals in the Lattice of Topologies

We discuss the question whether every finite interval in the lattice of all topologies on some set is isomorphic to an interval in the lattice of all topologies on a finite set – or, equivalently, whether the finite intervals in lattices of topologies are, up to isomorphism, exactly the duals of finite intervals in lattices of quasiorders. The answer to this question is in the affirmative at least for finite atomistic lattices. Applying recent results about intervals in lattices of quasiorders, we see that, for example, the five-element modular but non-distributive lattice cannot be an interval in the lattice of topologies. We show that a finite lattice whose greatest element is the join of two atoms is an interval of T ₀-topologies iff it is the four-element Boolean lattice or the five-element non-modular lattice. But only the first of these two selfdual lattices is an interval of orders because order intervals are known to be dually locally distributive.     

Finite Intervals in the Lattice of Topologies

We prove a conjecture of Reinhold: that a finite lattice is isomorphic to an interval in the lattice of topologies on some set if and only if it is isomorphic to an interval in the lattice of topologies on a finite set.     

Diagonalization of compact operators in Hilbert modules over finiteW*-algebras

It is known that a continuous family of compact self-adjoint operators can be diagonalized pointwise. One can consider this fact as a possibility of diagonalization of the compact operators on Hilbert modules over a commutativeW*-algebra. The aim of the present paper is to generalize this fact to a finiteW*-algebraA not necessarily commutative. We prove that for a compact operatorK acting on the right HilbertA-moduleH* _A dual toH _A under slight restrictions one can find a set of eigenvectorsx _i H* _A and a non-increasing sequence of eigenvalues _i A such thatK x _i=x _{i i} and the selfdual HilbertA-module generated by these eigenvectors is the wholeH* _A. As an application we consider the Schrödinger operator in a magnetic field with irrational magnetic flow as an operator acting on a Hilbert module over the irrational rotation algebraA and discuss the possibility of its diagonalization.     

Diagonalization of compact operators on Hilbert modules overC *-algebras of real rank zero

The classical Hilbert-Schmidt theorem can be extended to compact operators on HilbertA-modules overW ^*-algebras of finite type; i.e., with minor restrictions, compact operators onH* _A can be diagonalized overA. We show that ifB is a weakly denseC ^*-subalgebra ofA with real rank zero and if some additional condition holds, then the natural extension fromH _B toH* _A ⊃H _B of a compact operator can be diagonalized so that the diagonal elements belong to the originalC ^*-algebraB. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 865–870, December, 1997. Translated by O. V. Sipacheva     

On the Lattice of Antichains of Finite Intervals

Motivated by applications to information retrieval, we study the lattice of antichains of finite intervals of a locally finite, totally ordered set. Intervals are ordered by reverse inclusion; the order between antichains is induced by the lower set they generate. We discuss in general properties of such antichain completions; in particular, their connection with Alexandrov completions. We prove the existence of a unique, irredundant ∧-representation by ∧-irreducible elements, which makes it possible to write the relative pseudo-complement in closed form. We also discuss in detail properties of additional interesting operators used in information retrieval. Finally, we give a formula for the rank of an element and for the height of the lattice.     

On the Hilbert Transform in Bergman Space

In this paper we describe the image of the Hilbert transform operator for Bergman space.     

一类高振荡Hilbert变换的计算

Hilbert变换在信号处理与医学图像处理中都有着广泛的应用,但是对于一般的含有高振荡因子且在积分区阍中包含奇异值的‰变换往往处理起来较为困难,本文提出了一种基于等距节点插值的高效计算方法。     

Approximation of the Hilbert Transform on the real semiaxis using Laguerre zeros

The authors propose two new algorithms for the computation of Cauchy principal value integrals on the real semiaxis. The proposed quadrature rules use zeros of Laguerre polynomials. Theoretical error estimates are proved and some numerical examples are showed.     

Approximation of the Hilbert Transform on the real line using Hermite zeros

The authors study the Hilbert Transform on the real line. They introduce some polynomial approximations and some algorithms for its numerical evaluation. Error estimates in uniform norm are given.

     

基于Hilbert变换的LFM检测与估计

LFM(线性调频)信号是一类重要的非平稳信号,其完全被初始频率和调频斜率两个参量表征,而LFM信号的检测与估计问题是信号处理中最为重要的研究热点之一.由于调频信号在时频平面内有较好的聚集性,通常使用时频分析的方法对其进行检测和估计.线性正则变换是经典时频分布的广义形式,对LFM信号具有很好的能量聚集特性,在现有的线性正则域Hilbert变换的基础上,提出了一种不需要谱峰搜索而快速检测LFM信号和估计其参数的方法,并且通过仿真实例验证了所提出方法的优越性.     

A Hilbert Transform for Matrix Functions on Fractal Domains

We consider H?lder continuous circulant (2 × 2) matrix functions G¹₂{{\bf G}^1_2} defined on the fractal boundary Γ of a Jordan domain Ω in \mathbbR²ⁿ{\mathbb{R}^{2n}}. The main goal is to establish a Hilbert transform for such functions, within the framework of Hermitian Clifford analysis. This is a higher dimensional function theory centered around the simultaneous null solutions of two first order vector valued differential operators, called Hermitian Dirac operators. In Brackx et al. (Bull Braz Math Soc 40(3): 395–416, 2009) a Hermitian Cauchy integral was constructed by means of a matrix approach using circulant (2 × 2) matrix functions, from which a Hilbert transform was derived in Brackx et al. (J Math Anal Appl 344: 1068–1078, 2008) for the case of domains with smooth boundary. However, crucial parts of the method are not extendable to the case where the boundary of the considered domain is fractal. At present we propose an alternative approach which will enable us to define a new Hermitian Hilbert transform in that case. As a consequence, we give necessary and sufficient conditions for the Hermitian monogenicity of a circulant matrix function G¹₂{{\bf G}^1_2} in the interior and exterior of Ω, in terms of its boundary value g¹₂=G¹₂|_G{{\bf g}^1_2={\bf G}^1_2|_\Gamma}, extending in this way also results of Abreu Blaya et al. (Bound. Value Probl. 2008: 2008) (article ID 425256), (article ID 385874), where Γ is required to be Ahlfors–David regular.     

Periodicity of the Spectrum of a Finite Union of Intervals

A set Ω, of Lebesgue measure 1, in the real line is called spectral if there is a set Λ of real numbers such that the exponential functions e _λ (x)=exp (2πiλx), λ∈Λ, form a complete orthonormal system on L ²(Ω). Such a set Λ is called a spectrum of Ω. In this note we present a simplified proof of the fact that any spectrum Λ of a set Ω which is finite union of intervals must be periodic. The original proof is due to Bose and Madan.     

Characterization of the Weak-Type Boundedness of the Hilbert Transform on Weighted Lorentz Spaces

We characterize the weak-type boundedness of the Hilbert transform H on weighted Lorentz spaces $\varLambda^{p}_{u}(w)$ , with p>0, in terms of some geometric conditions on the weights u and w and the weak-type boundedness of the Hardy–Littlewood maximal operator on the same spaces. Our results recover simultaneously the theory of the boundedness of H on weighted Lebesgue spaces L ^p (u) and Muckenhoupt weights A _p , and the theory on classical Lorentz spaces Λ ^p (w) and Ariño-Muckenhoupt weights B _p .