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.

     

Simple proofs of the Bedrosian equality for the Hilbert transform

Two simple proofs of Bedrosian theorem are given.      

The circular Bedrosian identity for translation‐invariant operators: existence and characterization

The analytic signal method via the circular Hilbert transform is a critical tool in the time–frequency analysis of signals of finite duration. The circular Bedrosian identity is of major theoretical and practical value in the method. The identity holds whenever the Fourier coefficients of f,g∈L²([?π,π]) are respectively supported on A = [?n,m] and for some non‐negative integers 0≤n,m≤+∞. In this note, we investigate the existence of such an identity for a general‐bounded linear translation‐invariant operator on L²([?π,π]^d) and for general support sets . We give an insightful geometric characterization of the support sets for the existence. In addition, we find all the support sets for the partial Hilbert transforms. Copyright © 2015 John Wiley & Sons, Ltd.     

Hilbert transform of

There are two general ways to evaluate the Hilbert transform of a function of real variable . We can extend to a harmonic function in the upper half plane by the Poisson integral formula. Non-tangential limit of its harmonic conjugate exists almost everywhere and is defined to be the Hilbert transform of . There is also a singular integral formula for the Hilbert transform of . It is fairly difficult to directly evaluate the Hilbert transform of . In this paper we give an explicit formula for the Hilbert transform of , where is a function in the Cartwright class.

     

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)     

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 Hilbert transform and Hermite functions: A real variable proof of the -isometry

A proof of the fact that the Hilbert transform can be extended as an isometry to L² is obtained by real variable methods using the Hermite functions.     

A matrix Hilbert transform in Hermitean Clifford analysis

Orthogonal Clifford analysis is a higher dimensional function theory offering both a generalization of complex analysis in the plane and a refinement of classical harmonic analysis. During the last years, Hermitean Clifford analysis has emerged as a new and successful branch of it, offering yet a refinement of the orthogonal case. Recently in [F. Brackx, B. De Knock, H. De Schepper, D. Peña Peña, F. Sommen, submitted for publication], a Hermitean Cauchy integral was constructed in the framework of circulant (2×2) matrix functions. In the present paper, a new Hermitean Hilbert transform is introduced, arising naturally as part of the non-tangential boundary limits of that Hermitean Cauchy integral. The resulting matrix operator is shown to satisfy properly adapted analogues of the characteristic properties of the Hilbert transform in classical analysis and orthogonal Clifford analysis.     

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.     

Lp$L^p$ bound for the Hilbert transform along variable non-flat curves

We prove the $L^p$ bound for the Hilbert transform along variable non-flat curves $(t,u(x){[t]}^{\alpha }+v(x){[t]}^{\beta })$, where α and β satisfy $\alpha \ne \beta ,\alpha \ne 1,\beta \ne 1$. Compared with the associated theorem in the work (Guo et al. Proc. Lond. Math. Soc. 2017) investigating the case $\alpha =\beta \ne 1$, our result is more general while the proof is more involved. To achieve our goal, we divide the frequency of the objective function into three cases and take different strategies to control these cases. Furthermore, we need to introduce a “short” shift maximal function ${\mathbf{M}}^{[n]}$ to establish some pointwise estimates.     

Maximal theorems for the directional Hilbert transform on the plane

For a Schwartz function on the plane and a non-zero define the Hilbert transform of in the direction to be

p.v.

Let be a Schwartz function with frequency support in the annulus , and . We prove that the maximal operator maps into weak , and into for . The estimate is sharp. The method of proof is based upon techniques related to the pointwise convergence of Fourier series. Indeed, our main theorem implies this result on Fourier series.

     

On a hybrid of bilinear Hilbert transform and paraproduct

We introduce a class of tri-linear operators that combine features of the bilinear Hilbert transform and paraproduct. For two instances of these operators, we prove boundedness property in a large range D = {(p1,p2,p_3) ∈ R~3 : 1 p1,p2 ∞,1/(p1)+ 1/(p2)3/2,1 p3 ∞}.     

Integral inequalities for the Hilbert transform applied to a nonlocal transport equation

We prove several weighted inequalities involving the Hilbert transform of a function f(x) and its derivative. One of those inequalities,

is used to show finite time blow-up for a transport equation with nonlocal velocity.     

An estimate for weighted Hilbert transform via square functions

We show that the norm of the Hilbert transform as an operator on the weighted space is bounded by a constant multiple of the power of the constant of , in other words by . We also give a short proof for sharp upper and lower bounds for the dyadic square function.

     

A version of Burkholder's theorem for operator-weighted spaces

Let be an operator weight, i.e. a weight function taking values in the bounded linear operators on a Hilbert space . We prove that if the dyadic martingale transforms are uniformly bounded on for each dyadic grid in , then the Hilbert transform is bounded on as well, thus providing an analogue of Burkholder's theorem for operator-weighted -spaces. We also give a short new proof of Burkholder's theorem itself. Our proof is based on the decomposition of the Hilbert transform into ``dyadic shifts'.

     

A novel method for computing the Hilbert transform with Haar multiresolution approximation

In this paper, an algorithm for computing the Hilbert transform based on the Haar multiresolution approximation is proposed and the L²$L^2$-error is estimated. Experimental results show that it outperforms the library function ‘hilbert’ in Matlab (The MathWorks, Inc. 1994–2007). Finally it is applied to compute the instantaneous phase of signals approximately and is compared with three existing methods.     

Investigations on the approximability and computability of the Hilbert transform with applications

It was recently shown that on a large class of important Banach spaces there exist no linear methods which are able to approximate the Hilbert transform from samples of the given function. This implies that there is no linear algorithm for calculating the Hilbert transform which can be implemented on a digital computer and which converges for all functions from the corresponding Banach spaces. The present paper develops a much more general framework which also includes non-linear approximation methods. All algorithms within this framework have only to satisfy an axiom which guarantees the computability of the algorithm based on given samples of the function. The paper investigates whether there exists an algorithm within this general framework which converges to the Hilbert transform for all functions in these Banach spaces. It is shown that non-linear methods give actually no improvement over linear methods. Moreover, the paper discusses some consequences regarding the Turing computability of the Hilbert transform and the existence of computational bases in Banach spaces.     

Effective numerical evaluation of the double Hilbert transform

In this paper, we propose two methods to compute the double Hilbert transform of periodic functions. First, we establish the quadratic formula of trigonometric interpolation type for double Hilbert transform and obtain an estimation of the remainder. We call this method 2D mechanical quadrature method (2D-MQM). Numerical experiments show that 2D-MQM outperforms the library function “hilbert” in Matlab when the values of the functions being handled are very large or approach to infinity. Second, we propose a complex analytic method to calculate the double Hilbert transform, which is based on the 2D adaptive Fourier decomposition, and the method is called as 2D-HAFD. In contrast to the pointwise approximation, 2D-HAFD provides explicit rational functional approximations and is valid for all signals of finite energy.     

On approximating the finite Hilbert transform and applications in numerical integration

In this study, approximating the finite Hilbert transform are given for absolutely continuous mappings. Then, some numerical experiments for the obtained approximation are also presented.