scholarly journals Generalized multidimensional Hilbert transforms in Clifford analysis

2006 ◽  
Vol 2006 ◽  
pp. 1-19 ◽  
Author(s):  
Fred Brackx ◽  
Bram De Knock ◽  
Hennie De Schepper
Keyword(s):  
Sobolev Spaces ◽  
Hilbert Transform ◽  
Clifford Analysis ◽  
Hilbert Transforms ◽  
Singular Operators ◽  
The Hilbert Transform

Two specific generalizations of the multidimensional Hilbert transform in Clifford analysis are constructed. It is shown that though in each of these generalizations some traditional properties of the Hilbert transform are inevitably lost, new bounded singular operators emerge on Hilbert or Sobolev spaces ofL2-functions.

Toward a three‐dimensional automatic interpretation of potential field data via generalized Hilbert transforms: Fundamental relations

Geophysics ◽  
1984 ◽  
Vol 49 (6) ◽  
pp. 780-786 ◽  
Author(s):  
Misac N. Nabighian
Keyword(s):  
Hilbert Transform ◽  
Potential Field ◽  
Field Data ◽  
Three Dimensional ◽  
Analytic Signal ◽  
Three Dimensions ◽  
Hilbert Transforms ◽  
Potential Field Data ◽  
Automatic Interpretation ◽  
The Hilbert Transform

The paper extends to three dimensions (3-D) the two‐dimensional (2-D) Hilbert transform relations between potential field components. For the 3-D case, it is shown that the Hilbert transform is composed of two parts, with one part acting on the X component and one part on the Y component. As for the previously developed 2-D case, it is shown that in 3-D the vertical and horizontal derivatives are the Hilbert transforms of each other. The 2-D Cauchy‐Riemann relations between a potential function and its Hilbert transform are generalized for the 3-D case. Finally, the previously developed concept of analytic signal in 2-D can be extended to 3-D as a first step toward the development of an automatic interpretation technique for potential field data.

Hilbert transforms and almost periodic functions

1960 ◽  
Vol 56 (4) ◽  
pp. 354-366 ◽  
Author(s):  
J. Cossar
Keyword(s):  
Hilbert Transform ◽  
Sufficient Conditions ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Hilbert Transforms ◽  
Cauchy Principal ◽  
Principal Value ◽  
Image Position ◽  
The Hilbert Transform

The Hilbert transform, Hf, of a function f is defined by Hf = g, whereP denoting the Cauchy principal value and the integral being assumed to exist in some sense. When f is suitably restricted, Hf exists andIn the first part of Theorem 1 sufficient conditions are given for the validity of (1·2) rather more general than those of Wood ((6), p. 31). The present proof is based on the well-known condition of Riesz for the validity of (1·2), namely, that f is Lp(−∞, ∞) for some p > 1, and on the ‘Parseval’ relation (Lemma 3, (b)), which was used in a similar way by Hardy ((3), p. 110).

scholarly journals On the Cauchy problems associated to a ZK-KP-type family equations with a transversal fractional dispersion

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jorge Morales Paredes ◽  
Félix Humberto Soriano Méndez
Keyword(s):  
Fractional Derivative ◽  
Sobolev Spaces ◽  
Hilbert Transform ◽  
Well Posedness ◽  
Cauchy Problems ◽  
Anisotropic Sobolev Spaces ◽  
The Hilbert Transform

<p style='text-indent:20px;'>In this paper we examine the well-posedness and ill-posedeness of the Cauchy problems associated with a family of equations of ZK-KP-type</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} u_{t} = u_{xxx}-\mathscr{H}D_{x}^{\alpha}u_{yy}+uu_{x}, \cr u(0) = \psi \in Z \end{cases} $\end{document} </tex-math> </disp-formula></p><p style='text-indent:20px;'>in anisotropic Sobolev spaces, where <inline-formula><tex-math id="M1">\begin{document}$ 1\le \alpha \le 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \mathscr{H} $\end{document}</tex-math></inline-formula> is the Hilbert transform and <inline-formula><tex-math id="M3">\begin{document}$ D_{x}^{\alpha} $\end{document}</tex-math></inline-formula> is the fractional derivative, both with respect to <inline-formula><tex-math id="M4">\begin{document}$ x $\end{document}</tex-math></inline-formula>.</p>

On: “Interpretation of magnetic bodies using Hilbert transforms” by N. L. Mohan, N. Sundararajan, and S. V. Seshagiri Rao (GEOPHYSICS, 47, 376–387, March, 1982).

Geophysics ◽  
1985 ◽  
Vol 50 (1) ◽  
pp. 168-168
Author(s):  
Rob Pauls
Keyword(s):  
Data Processing ◽  
Hilbert Transform ◽  
Geophysical Data ◽  
Hilbert Transforms ◽  
Magnetic Interpretation ◽  
The Hilbert Transform

The Fourier and Hilbert transform techniques have played extremely important roles in geophysical data processing. We were naturally very interested to find the paper by Mohan et al. which described a possible application of the Hilbert transform approach in magnetic interpretation. However, we found some very serious problems in the paper.

On: “Interpretation of two dimensional magnetic bodies using Hilbert transforms,” by N. L. Mohan, N. Sundararajan, and S. V. Seshagiri Rao (March 1982 GEOPHYSICS, 52, p. 376–387)

Geophysics ◽  
1997 ◽  
Vol 62 (2) ◽  
pp. 690-691
Author(s):  
B. N. P. Agarwal
Keyword(s):  
Magnetic Field ◽  
Hilbert Transform ◽  
Two Dimensional ◽  
Hilbert Transforms ◽  
The Hilbert Transform

While going through some of the publications (Mohan and Babu, 1995), I became interested in the work of Mohan et al. (1982) which proposed a technique for interpretation of magnetic field anomalies over different geometrical sources using the Hilbert transform (HT). Before I put forward my observations, it would be appropriate to look into some important properties of HT (Whalen, 1971, pages 63 and 69.)

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.

On the Essential Norm for the Hilbert Transforms in 𝐿𝑝(𝑥) Spaces

2008 ◽  
Vol 15 (2) ◽  
pp. 209-223
Author(s):  
Muhammad Asif ◽  
Alexander Meskhi
Keyword(s):  
Hilbert Transform ◽  
Variable Exponent ◽  
Essential Norm ◽  
Lebesgue Spaces ◽  
Hilbert Transforms ◽  
Weighted Lebesgue Spaces ◽  
The Hilbert Transform

Abstract The essential norm of the Hilbert transform acting in weighted Lebesgue spaces with variable exponent is estimated from below.

On: Interpretation of some two‐dimensional magnetic bodies using Hilbert transforms (GEOPHYSICS, v. 47, p. 376–387).

Geophysics ◽  
1983 ◽  
Vol 48 (2) ◽  
pp. 248-248
Author(s):  
J. Roth
Keyword(s):  
Hilbert Transform ◽  
Potential Field ◽  
Two Dimensional ◽  
Hilbert Transforms ◽  
The Hilbert Transform

The above‐cited paper usefully examines and extends the application of the Hilbert transform to potential field interpretation. However, the authors’ terse mention of Nabighian’s paper (Geophysics, 1972) fails to characterize adequately and acknowledge his original insights and contributions to the Hilbert transform presented in that paper. It is surprising as well that none of the reviewers and/or editors saw fit to rectify this undeserved omission.

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