Computing the Hilbert transform using biorthogonal spline wavelets

In this paper, we summarize some facts on spline wavelets, analyze the Hilbert transform of these wavelets on the real line and on the unit circle, describe an algorithm for computing the Hilbert transform on uniform grids, and report on some test calculations.     

On the Hilbert transform

Для любой функцииf∈L ^p (|x|^α dx, гд е p >1, аа — любое действительное числ о, найдены необходимые и достаточные услови я для того, чтобы ее преобразова ние ГильбертаHf $$Hf(x) = \frac{1}{\pi }\mathop \smallint \limits_{ - \infty }^\infty (x - t)^{ - 1} f(t)dt, x \in ( - \infty ,\infty )$$ , также принадлежалоL ^p (|x|)^α dx.L ^p (|x|)^α dx — класс измеримых фу нкцийf определенных на (- ∞, ∞) и удовлетворяющих условию $$\mathop \smallint \limits_{ - \infty }^\infty |f(x)|^p |x|^\alpha dx< \infty$$ . Ниже сформулированы наиболее интересные случаи: Теорема 1. Теорема 2.      

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.     

Fourier transform versus Hilbert transform

Certain relations between the Fourier transform of a function and the Hilbert transform of its derivative are revealed. They concern the integrability/non-integrability of both transforms. Certain applications are discussed.     

On the convergence of the rotated one-sided ergodic Hilbert transform

Sufficient conditions have been given for the convergence in norm and a.e. of the ergodic Hilbert transform (Gaposhkin in Theory Probab Appl 41:247–264, 1996; Cohen and Lin in Characteristic functions, scattering functions and transfer functions, pp 77–98, Birkhäuser, Basel, 2009; Cuny in Ergod Theory Dyn Syst 29:1781–1788, 2009). Here we apply these conditions to the rotated ergodic Hilbert transform \({\sum_{n=1}^\infty \frac{\lambda^n}{n} T^nf}\) , where λ is a complex number of modulus 1. When T is a contraction in a Hilbert space, we show that the logarithmic Hausdorff dimension of the set of λ’s for which this series does not converge is at most 2 and give examples where this bound is attained.     

On theA-integral representation of the Hilbert transform and conjugate function

Рассматривается воп рос о представлении о ператора Гильберта и сопряжен ной функцииA-интегралом. Доказывается следую щая Теорема. Если ? - такая неотрицательная фун кция на [0, ∞), что х^?1?(х) монотонно не убывает на (0, ∞) и для н екоторого Н> 0 \(\mathop \smallint \limits_H^\infty \varphi ^{ - 1} (x)dx< \infty\) , а определенная на R функ ция fε?∩?(?), то почти всюду оператор Гильберта $$\tilde f(x) = - \frac{1}{\pi }(A)\mathop \smallint \limits_0^\infty \frac{{f(x + t) - f(x - t)}}{t}dt$$ . Из данной теоремы сле дует, что для функций и з ?^p, 1<р<#x221E;, оператор Гильберта и сопряженная функция представляютсяA-инте гралом. Что для функций из ?¹ п одобное утверждение неверно, показывает следующа я теорема. Теорема.Существует т акая суммируемая на R ф ункция f≧0, что почти всюду $$\mathop {\lim sup}\limits_{n \to \infty } \mathop \smallint \limits_0^\infty \left[ {\frac{{f(x + t) - f(x - t)}}{t}} \right]_n dt = \infty$$ .     

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.     

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 ∞}.     

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.     

Traces of the discrete Hilbert transform with quadratic phase

For the function $H:\mathbb{R}^2 \mapsto \mathbb{C}$ , $H: = (p.v.)\sum\nolimits_{n \in \mathbb{Z}\backslash \{ 0\} } {\tfrac{{\exp \left\{ {\pi i\left( {tn^2 + 2xn} \right)} \right\}}} {{2\pi in}}}$ of two real variables (t, x) ∈ ?², we study the uniform moduli of continuity and the variations of the restrictions H| _t and H| _x onto the lines parallel to the coordinate axes x = 0 and t = 0. Smoothness of such restrictions is primarily determined by the Diophantine approximation of the fixed parameter. Generalized (weak) variations are also studied, and it is shown in particular that sup _x w₄[H| _x ] < ∞ where w₄ denotes the weak quartic variation. Previously it was known that uniformly in the parameter t ∈ ?, the restriction H| _t is a function of bounded weak quadratic variation in the variable x, i.e., sup _t w₂[H| _t ] < ∞. The function H has multiple applications: in the study of the spectra of uniform convergence (P.L. Ul’yanov’s problem), in the incomplete Gaussian sums (where it plays the role of the generating function), in the partial differential equations of mathematical physics (in the Cauchy problem for the Schrödinger equation), and in quantum optics (Talbot’s phenomenon).     

Double Hilbert transform on

We introduce a space , where is the testing function space whose functions are infinitely differentiable and have bounded support, and is the space the double Hilbert transform acting on the testing function space. We prove that the double Hilbert transform is a homeomorphism from onto itself.     

A characterization of the Hilbert transform

In this note the Hilbert transform is characterized in terms of function algebras with respect to pointwise multiplication.

     

On the numerical inversion of the Laplace transform in reproducing kernel Hilbert spaces

In this work we propose a method for the numerical inversionof the Laplace transform in two reproducing kernel Hilbert spaces,based on the hypothesis that the recovering function is continuousand that the values at the ends of its range are known. Thesolution is given by a weighted linear combination of Jacobipolynomials whose coefficients are expressed in terms of theLaplace transform evaluated at equally spaced points. The effectivenessof the method is illustrated by the recovery of a number offunctions for the most part already proposed in the literature.     

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.

     

Distributional estimates for the bilinear Hilbert transform

We obtain size estimates for the distribution function of the bilinear Hilbert transform acting on a pair of characteristic functions of sets of finite measure, that yield exponential decay at infinity and blowup near zero to the power −2/3 (modulo some logarithmic factors). These results yield all known L^p bounds for the bilinear Hilbert transform and provide new restricted weak type endpoint estimates on L^p1 × L^p2 when either 1/p₁ + 1/p₂ = 3/2 or one of p₁, p₂ is equal to 1. As a consequence of this work we also obtain that the square root of the bilinear Hilbert transform of two characteristic functions is exponentially integrable over any compact set.     

Identity for a spectral function connected with the Hilbert transform

We give a new proof of the identity, known as the “sum rule” in the statistical quantum mechanics, for the integral over the real axis of a nonlinear combination of a function u(x) in a certain class and its Hilbert transform. Examples where the identity fails are given Bibliography: 5 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 36, 2007, pp. 23–28.