scholarly journals On the p-norm of the truncated Hilbert transform

1988 ◽  
Vol 38 (3) ◽  
pp. 413-420 ◽  
Author(s):  
W. McLean ◽  
D. Elliott
Keyword(s):  
Hilbert Transform ◽  
Positive Measure ◽  
Integral Operators ◽  
Finite Hilbert Transform ◽  
Measurable Set ◽  
Symmetry Properties ◽  
Lebesgue Measurable Set ◽  
The One ◽  
The Hilbert Transform

The p-norm of the Hilbert transform is the same as the p-norm of its truncation to any Lebesgue measurable set with strictly positive measure. This fact follows from two symmetry properties, the joint presence of which is essentially unique to the Hilbert transform. Our result applies, in particular, to the finite Hilbert transform taken over (−1, 1), and to the one-sided Hilbert transform taken over (0, ∞). A related weaker property holds for integral operators with Hardy kernels.

MAXIMAL SEMIGROUP SYMMETRY AND DISCRETE RIESZ TRANSFORMS

2015 ◽  
Vol 100 (2) ◽  
pp. 216-240
Author(s):  
TOSHIYUKI KOBAYASHI ◽  
ANDREAS NILSSON ◽  
FUMIHIRO SATO
Keyword(s):  
Hilbert Transform ◽  
Riesz Transform ◽  
Riesz Transforms ◽  
Dimensional Case ◽  
One Dimensional ◽  
Linearly Independent ◽  
The One ◽  
The Hilbert Transform ◽  
Multiplier Operators

We raise a question of whether the Riesz transform on $\mathbb{T}^{n}$ or $\mathbb{Z}^{n}$ is characterized by the ‘maximal semigroup symmetry’ that the transform satisfies. We prove that this is the case if and only if the dimension is one, two or a multiple of four. This generalizes a theorem of Edwards and Gaudry for the Hilbert transform on $\mathbb{T}$ and $\mathbb{Z}$ in the one-dimensional case, and extends a theorem of Stein for the Riesz transform on $\mathbb{R}^{n}$. Unlike the $\mathbb{R}^{n}$ case, we show that there exist infinitely many linearly independent multiplier operators that enjoy the same maximal semigroup symmetry as the Riesz transforms on $\mathbb{T}^{n}$ and $\mathbb{Z}^{n}$ if the dimension $n$ is greater than or equal to three and is not a multiple of four.

The finite Hilbert transform in weighted spaces

1996 ◽  
Vol 126 (6) ◽  
pp. 1157-1167 ◽  
Author(s):  
Kari Astala ◽  
Lassi Päivärinta ◽  
Eero Saksman
Keyword(s):  
Hilbert Transform ◽  
Weighted Spaces ◽  
Logarithmic Kernel ◽  
Finite Hilbert Transform ◽  
Half Axis ◽  
The Hilbert Transform

The mapping properties of the finite Hilbert-transform (respectively the Hilbert transform on the half axis) are studied. Invertibility, surjectivity, injectivity and bounded ness from below of the transform are characterised in general weighted spaces. The results are applied to the restriction of the operator with logarithmic kernel.

Infrared intensities of liquids. IX. The Kramers–Kronig transform, and its approximation by the finite Hilbert transform via fast Fourier transforms

1992 ◽  
Vol 70 (2) ◽  
pp. 520-531 ◽  
Author(s):  
John E. Bertie ◽  
Shuliang L. Zhang
Keyword(s):  
Hilbert Transform ◽  
Fourier Transforms ◽  
Integral Transforms ◽  
Data File ◽  
Refractive Indices ◽  
Simple Method ◽  
K Value ◽  
Finite Hilbert Transform ◽  
Infrared Intensities ◽  
The Hilbert Transform

It is well known that the infinite Kramers–Kronig transform is equivalent to the infinite Hilbert transform, which is equivalent to the allied Fourier integrals. The Hilbert transform can thus be implemented using fast Fourier transform routines. Such implementation is usually some 60 times faster than the Kramers–Kronig transform for a data file containing about 7 points. This paper reports that, for transformations between the real and imaginary refractive indices, [Formula: see text] and [Formula: see text] in [Formula: see text], the FFT-based Hilbert transform can be much less accurate than, or as accurate as, the Kramers–Kronig transform, depending on the algorithm used. The Kramers–Kronig transform, incorporating Mclaurin's formula for finding the principal value of the integral, transforms [Formula: see text] spectra into [Formula: see text] spectra that are accurate to about 0.05%. Some Hilbert transform algorithms in the literature yield only about 4% accuracy. The BZ algorithm for the Hilbert transform is presented, for use on a laboratory computer running under DOS, that yields [Formula: see text] spectra accurate to 0.05%. For the transform from [Formula: see text] to [Formula: see text], the BZ algorithm gives [Formula: see text] accurate to about −0.2% of the largest k value in the spectrum. This compares with an accuracy of 0.5% for the Kramers–Kronig transform. In cases where the [Formula: see text] spectrum is truncated at low wavenumbers, a simple method is presented that improves by a factor of ~10 the accuracy at low wavenumber of the [Formula: see text] spectrum obtained by Hilbert or Kramers–Kronig transforms of the [Formula: see text] spectrum. Keywords: infrared intensities, complex refractive indices, Kramers–Kronig transform, Hilbert transform, optical constants.

scholarly journals Some remarks on Vainikko integral operators in BV type spaces

2020 ◽  
Vol 13 (4) ◽  
pp. 555-565
Author(s):  
Laura Angeloni ◽  
Jürgen Appell ◽  
Simon Reinwand
Keyword(s):  
Hilbert Transform ◽  
Integral Operators ◽  
Hardy Operator ◽  
Norm Estimates ◽  
Special Cases ◽  
Type Spaces ◽  
The Hilbert Transform

AbstractIn this paper we study Vainikko integral operators which are similar to so-called cordial integral operators and contain the classical Hardy operator, the Schur operator, and the Hilbert transform as special cases. For such operators we obtain norm estimates and equalities, mainly in BV type spaces in the sense of Jordan, Wiener, Riesz, and Waterman. Several examples are also discussed.

scholarly journals The Hilbert Transform and Orthogonal Martingales in Banach Spaces

2019 ◽  
Author(s):  
Adam Osękowski ◽  
Ivan Yaroslavtsev
Keyword(s):  
Banach Space ◽  
Singular Integral ◽  
Hilbert Transform ◽  
Real Line ◽  
Harmonic Functions ◽  
Integral Operators ◽  
Continuous Functions ◽  
Hilbert Transforms ◽  
Discrete Hilbert Transform ◽  
The Hilbert Transform

Abstract Let $X$ be a given Banach space, and let $M$ and $N$ be two orthogonal $X$-valued local martingales such that $N$ is weakly differentially subordinate to $M$. The paper contains the proof of the estimate $\mathbb E \Psi (N_t) \leq C_{\Phi ,\Psi ,X} \mathbb E \Phi (M_t)$, $t\geq 0$, where $\Phi , \Psi :X \to \mathbb R_+$ are convex continuous functions and the least admissible constant $C_{\Phi ,\Psi ,X}$ coincides with the $\Phi ,\Psi $-norm of the periodic Hilbert transform. As a corollary, it is shown that the $\Phi ,\Psi $-norms of the periodic Hilbert transform, the Hilbert transform on the real line, and the discrete Hilbert transform are the same if $\Phi $ is symmetric. We also prove that under certain natural assumptions on $\Phi $ and $\Psi $, the condition $C_{\Phi ,\Psi ,X}<\infty $ yields the UMD property of the space $X$. As an application, we provide comparison of $L^p$-norms of the periodic Hilbert transform to Wiener and Paley–Walsh decoupling constants. We also study the norms of the periodic, nonperiodic, and discrete Hilbert transforms and present the corresponding estimates in the context of differentially subordinate harmonic functions and more general singular integral operators.

Quantitative continuation from a measurable set of solutions of elliptic equations

2000 ◽  
Vol 130 (4) ◽  
pp. 909-923 ◽  
Author(s):  
S. Vessella
Keyword(s):  
Elliptic Equation ◽  
Elliptic Equations ◽  
Positive Measure ◽  
Symmetric Matrix ◽  
Measurable Set ◽  
Connected Set ◽  
Lebesgue Measurable Set

Consider an open bounded connected set Ω in Rn and a Lebesgue measurable set E ⊂⊂ Ω of positive measure. Let u be a solution of the strictly elliptic equation Di (aij Dj u) = 0 in Ω, where aij ∈ C0, 1 (Ω̄) and {aij} is a symmetric matrix. Assume that |u| ≤ ε in E. We quantify the propagation of smallness of u in Ω.

Model of multicomponent narrow-band periodical non-stationary random signal

2020 ◽  
Vol 2020 (48) ◽  
pp. 17-24
Author(s):  
I.M. Javorskyj ◽  
◽  
R.M. Yuzefovych ◽  
P.R. Kurapov ◽  
◽  
...  
Keyword(s):  
Time Series ◽  
Real Time ◽  
Hilbert Transform ◽  
Spectral Properties ◽  
Narrow Band ◽  
Analytic Signal ◽  
Random Signal ◽  
Hilbert Transformation ◽  
The Hilbert Transform

The correlation and spectral properties of a multicomponent narrowband periodical non-stationary random signal (PNRS) and its Hilbert transformation are considered. It is shown that multicomponent narrowband PNRS differ from the monocomponent signal. This difference is caused by correlation of the quadratures for the different carrier harmonics. Such features of the analytic signal must be taken into account when we use the Hilbert transform for the analysis of real time series.

A Bayesian View on the Hilbert Transform and the Kramers-Kronig Transform of Electrochemical Impedance Data: Probabilistic Estimates and Quality Scores

2020 ◽  
Author(s):  
Jiapeng Liu ◽  
Ting Hei Wan ◽  
Francesco Ciucci
Keyword(s):  
Power Generation ◽  
Electrochemical Impedance Spectroscopy ◽  
Hilbert Transform ◽  
Electrochemical Impedance ◽  
The Self ◽  
Impedance Data ◽  
The Hilbert Transform ◽  
Validation Tests ◽  
Self Consistency ◽  
Mathematical Relations

<p>Electrochemical impedance spectroscopy (EIS) is one of the most widely used experimental tools in electrochemistry and has applications ranging from energy storage and power generation to medicine. Considering the broad applicability of the EIS technique, it is critical to validate the EIS data against the Hilbert transform (HT) or, equivalently, the Kramers–Kronig relations. These mathematical relations allow one to assess the self-consistency of obtained spectra. However, the use of validation tests is still uncommon. In the present article, we aim at bridging this gap by reformulating the HT under a Bayesian framework. In particular, we developed the Bayesian Hilbert transform (BHT) method that interprets the HT probabilistic. Leveraging the BHT, we proposed several scores that provide quick metrics for the evaluation of the EIS data quality.<br></p>

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