On a Result of Telyakovskii and Multiple Hilbert Transforms with Polynomial Phases

In this paper, we prove a multiple analog of the theorem proved by Arkhipov and the author in 1987, which provides an estimate for the discrete Hilbert transform with polynomial phase. For the linear case, the corresponding estimates of the sum of multiple trigonometric series was proved by Telyakovskii.     

Characterizing the Hilbert transform by the Bedrosian theorem

It is proved that a bounded linear translation invariant operator on L²(Rd) satisfies the Bedrosian theorem if and only if it is a linear combination of the compositions of the partial Hilbert transforms and the identity operator. This observation justifies a definition of multidimensional analytic signals in the papers [T. Bulow, G. Sommer, Hypercomplex signals—a novel extension of the analytic signal to the multidimensional case, IEEE Trans. Signal Process. 49 (2001) 2844-2852] and [S.L. Hahn, Multidimensional complex signals with single-orthant spectra, Proc. IEEE 80 (1992) 1287-1300].     

A discrete Fourier transform based on Simpson's rule

Fourier analysis plays a vital role in the analysis of continuous‐time signals. In many cases, we are forced to approximate the Fourier coefficients based on a sampling of the time signal. Hence, the need for a discrete transformation into the frequency domain giving rise to the classical discrete Fourier transform. In this paper, we present a transformation that arises naturally if one approximates the Fourier coefficients of a continuous‐time signal numerically using the Simpson quadrature rule. This results in a decomposition of the discrete signal into two sequences of equal length. We show that the periodic discrete time signal can be reconstructed completely from its discrete spectrum using an inverse transform. We also present many properties satisfied by this transform. Copyright © 2012 John Wiley & Sons, Ltd.     

Sampling theorem and multi-scale spectrum based on non-linear Fourier atoms

This study concerns some new developments of unit analytic signals with non-linear phase. It includes ladder-shaped filter, generalized Sinc function based on non-linear Fourier atoms, generalized sampling theorem for non-bandlimited signals and the notion of multi-scale spectrum for discrete sequences. We first introduce the ladder-shaped filter and show that the impulse response of its corresponding linear time-shift invariant system is the generalized Sinc function as a product of periodic Poisson kernel and Sinc function. Secondly, we establish a Shannon-type sampling theorem based on generalized Sinc function for this type of non-bandlimited signal. We further prove that this type of signal may be holomorphically extended to strips in the complex plane containing the real axis. Finally, we introduce the notion of multi-scale spectrums for discrete sequences and develop the related fast algorithm.     

Properties of the Simpson discrete Fourier transform

Abstract

The Simpson discrete Fourier transform (SDFT) and its inverse are transformations relating the time and frequency domains. In this paper we state and prove the important properties of shift, circular convolution, conjugation, time reversal and Plancherel's theorem. In addition, we provide an alternative representation of the duality property.     

A distribution space for Hilbert transform and its applications

In this paper, a new distribution space is constructed and the definition of the classical Hilbert transform is extended to it. It is shown that is the biggest subspace of on which the extended Hilbert transform is a homeomorphism and both the classical Hilbert transform for L ^p functions and the circular Hilbert transform for periodic functions are special cases of the extension. Some characterizations of the space are given and a class of useful nonlinear phase signals is shown to be in . Finally, the applications of the extended Hilbert transform are discussed. This work was supported by the National Natural Science Foundation of China (Grant Nos. 60475042, 10631080)     

The Minimal Measurement Number Problem in Phase Retrieval: A Review of Recent Developments

Phase retrieval is to recover the signals from phaseless measurements which is raised in many areas. A fundamental problem in phase retrieval is to determine the minimal measurement number $m$ so that one can recover $d$-dimensional signals from $m$ phaseless measurements. This problem attracts much attention of experts from different areas. In this paper, we review the recent development on the minimal measurement number and also raise many interesting open questions.     

On the interplay between the Hilbert transform and conjugate harmonic functions

As is well‐known, there is a close and well‐defined connection between the notions of Hilbert transform and of conjugate harmonic functions in the context of the complex plane. This holds e.g. in the case of the Hilbert transform on the real line, which is linked to conjugate harmonicity in the upper (or lower) half plane. It also can be rephrased when dealing with the Hilbert transform on the boundary of a simply connected domain related to conjugate harmonics in its interior (or exterior). In this paper, we extend these principles to higher dimensional space, more specifically, in a Clifford analysis setting. We will show that the intimate relation between both concepts remains, however giving rise to a range of possibilities for the definition of either new Hilbert‐like transforms, or specific notions of conjugate harmonicity. Copyright © 2006 John Wiley & Sons, Ltd.     

The Bedrosian identity for the Hilbert transform of product functions

We investigate a necessary and sufficient condition which ensures validity of the Bedrosian identity for the Hilbert transform of a product function . Convenient sufficient conditions are presented, which cover the classical Bedrosian theorem and provide us with new insightful information.

     

Approximate Controllability of a Stochastic Wave Equation

In this paper we discuss the controllability of a wave equation with random noise. Our main tools are the Ito representation theorem and an adaptation of the Hilbert uniqueness method for the exact controllability of deterministic equations.     

Approximate Controllability of a Stochastic Wave Equation

In this paper we discuss the controllability of a wave equation with random noise. Our main tools are the Ito representation theorem and an adaptation of the Hilbert uniqueness method for the exact controllability of deterministic equations.     

Phaseless Sampling and Reconstruction of Real-Valued Signals in Shift-Invariant Spaces

Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. In this paper, we consider phaseless sampling and reconstruction of real-valued signals in a high-dimensional shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. The determination of a signal in a shift-invariant space, up to a sign, by its magnitude measurements on the whole Euclidean space has been shown in the literature to be equivalent to its nonseparability. In this paper, we introduce an undirected graph associated with the signal in a shift-invariant space and use connectivity of the graph to characterize nonseparability of the signal. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that nonseparable signals in the shift-invariant space can be reconstructed in a stable way from their phaseless samples taken on that set. In this paper, we also propose a reconstruction algorithm which provides an approximation to the original signal when its noisy phaseless samples are available only. Finally, numerical simulations are performed to demonstrate the robustness of the proposed algorithm to reconstruct box spline signals from their noisy phaseless samples.

     

判定P_n(Γ)的Hilbert基的一个充要

设Γ是一作用在R^n上的紧李群，P_n(Γ)是Γ不变的多项式芽构成的环. Hilbert-Weyl定理证明了对于P_n(Γ)总存在一组由Γ不变的齐次多项式芽组成的Hilbert基. 然而，如何从Γ不变的齐次多项式芽中选出一组Hilbert基？如何判定Γ不变的齐次多项式芽的一个有限集就是P_n(Γ)的一组Hilbert基？该文借助于Noether环和不变积分的某些基本性质以及奇点理论的有关定理，证明了判定P_n(Γ)的Hilbert基的一个充要条件. 这对某些P_n(Γ)提供了计算一组Hilbert基的新途径.     

Band limited functions on quantum graphs

The notion of band limited functions is introduced on a quantum graph. The main results of the paper are a uniqueness theorem and a reconstruction algorithm of such functions from discrete sets of values. It turns out that some of our band limited functions can have compact supports and their frequencies can be localized on the ``time" side. It opens an opportunity to consider signals of a variable band width and to develop a sampling theory with variable rate of sampling.

     

Sampling in a Hilbert space

An analog of the Whittaker-Shannon-Kotel'nikov sampling theorem is derived for functions with values in a separable Hilbert space. The proof uses the concept of frames and frame operators in a Hilbert space. One of the consequences of this theorem is that it allows us to derive sampling theorems associated with boundary-value problems and some homogeneous integral equations, which in turn gives us a generalization of another sampling theorem by Kramer.

     

Holder Norm Estimate for a Hilbert Transform in Hermitean Clifford Analysis

A Hilbert transform for H?lder continuous circulant (2 × 2) matrix functions, on the d-summable (or fractal) boundary Γ of a Jordan domain Ω in ?²ⁿ , has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the H?lder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the H?lder exponents, the diameter of Γ and a specific d-sum (d > d) of the Whitney decomposition of Ω. The result is shown to include the case of a more standard Hilbert transform for domains with left Ahlfors-David regular boundary.     

Stable Signal Recovery from Phaseless Measurements

The aim of this paper is to study the stability of the \(\ell _1\) minimization for the compressive phase retrieval and to extend the instance-optimality in compressed sensing to the real phase retrieval setting. We first show that \(m={\mathcal {O}}(k\log (N/k))\) measurements are enough to guarantee the \(\ell _1\) minimization to recover k-sparse signals stably provided the measurement matrix A satisfies the strong RIP property. We second investigate the phaseless instance-optimality presenting a null space property of the measurement matrix A under which there exists a decoder \(\Delta \) so that the phaseless instance-optimality holds. We use the result to study the phaseless instance-optimality for the \(\ell _1\) norm. This builds a parallel for compressive phase retrieval with the classical compressive sensing.     

An adaptive sampling method for high‐dimensional shift‐invariant signals

In this paper, an adaptive method for sampling and reconstructing high‐dimensional shift‐invariant signals is proposed. First, the integrate‐and‐fire sampling scheme and an approximate reconstruction algorithm for one‐dimensional bandlimited signals are generalized to shift‐invariant signals. Then, a high‐dimensional shift‐invariant signal is reduced to be a sequence of one‐dimensional shift‐invariant signals along the trajectories parallel to some coordinate axis, which can be approximately reconstructed by the generalized integrate‐and‐fire sampling scheme. Finally, an approximate reconstruction for the high‐dimensional shift‐invariant signal is obtained by solving a series of stable linear systems of equations. The main result shows that the final reconstructed error is completely determined by the initial threshold in integrate‐and‐fire sampling scheme, which is generally very small. Copyright © 2017 John Wiley & Sons, Ltd.     

The finite Hilbert transform and weighted Sobolev spaces

The boundedness of the finite Hilbert transform operator on certain weighted L_p spaces is well known. We extend this result to give the boundedness of that operator on certain weighted Sobolev spaces. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Backward shift invariant subspaces with applications to band preserving and phase retrieval problems

The band preserving and phase retrieval problems have long been interested and studied. In this paper, we, for the first time, give solutions to these problems in terms of backward shift invariant subspaces. The backward shift method among other methods seems to be direct and natural. We show that a function , with , that makes the band of fg to be within that of f if and only if g divided by an inner function related to f, belongs to some backward shift invariant subspace in relation to f. By the construction of backward shift invariant space, the solution g is further explicitly represented through the span of the rational function system whose zeros are those of the Laplace transform of f. As an application, we also use the backward shift method to give a characterization for the solutions of the phase retrieval problem. Copyright © 2015 John Wiley & Sons, Ltd.