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Let M(u), H(u) be the maximal operator and Hilbert transform along the parabola (t, ut2). For U ⊂ (0, ∞) we consider Lp estimates for the maximal functions sup u∈U|M(u)f| and sup u∈U|H(u)f|, when 1 < p ≤ 2. The parabolas can be replaced by more general non-flat homogeneous curves.
Useful discrete-time signals and systems properties are introduced. This is followed by a brief review of the z-transform. Spectral analysis of seismic data and useful transforms are discussed. Signal analysis in the spectral or other domains is very important and assists in obtaining a better understanding of signals. Particularly when dealing with seismic data, it becomes almost standard to analyze seismic data sets in the 2-D frequency-wavenumber domain. Also, other discrete transforms such as the Radon transform are very useful for processing seismic data sets, which can be used, for example, for seismic wavefield decomposition as well as seismic multiple removal.
A classical result of Fatou gives that every bounded holomorphic function on the disc has radial limits for almost every point in the torus, and the limit function belongs to the Hardy space H_\infty of the torus. This property is no longer true when we consider vector-valued functions. The Banach spaces X for which this property is satisfied are said to have the analytic Radon-Nikodym property (ARNP). Some important equivalent reformulations of ARNP are studied in this chapter. Among others, X has ARNP if and only if each X-valued H_p- function f on the disc has radial limits almost everywhere on the torus (and not only H_\infty-functions). Even more, in this case each such f has non-tangential limits within any Stolz region. The basic tools are subharmonic functions and certain maximal inequalities. Finally, it is shown that if X has the ARNP, then every L_p of functions taking values in X with a finite measure also has ARNP.
Given a Banach space X, we consider Hardy spaces of X-valued functions on the infinite polytorus, Hardy spaces of X-valued Dirichlet series (defined as the image of the previous ones by the Bohr transform), and Hardy spaces of X-valued holomorphic functions on l_2 ∩ B_{c0}. The chapter is dedicated to study the interplay between these spaces. It is shown that the space of functions on the polytorus always forms a subspace of the one of holomorphic functions, and these two are isometrically isomorphic if and only if X has ARNP. Then the question arises of what do we find in the side of Dirichlet series when we look at the image of the Hardy space of holomorphic functions. This is also answered, showing that this consists of Dirichlet series for which all horizontal translations (those whose coefficients are (a_n/n^ε)) are in \mathcal{H}_p with uniformly bounded norms. Also, a version of the brothers Riesz theorem for vector-valued functions is given.
For each 1 ≤ p ≤ ∞, the Hardy space \mathcal{H}_p of Dirichlet series is defined as the image through the Bohr transform of the Hardy space of functions on the infinite-dimensional polytorus. The Dirichlet polynomials are dense in \mathcal{H}_p for every 1 ≤ p < ∞. For p=2 this coincides with the space of Dirichlet series whose coefficients are square-summable. A Dirichlet series with coefficients a_n belongs to\mathcal{H}_p if and only if the series with coefficients a_n/n^ε is in \mathcal{H}_p for every ε >0 and the norms are uniformly bounded. In this case, the series is the limit as ε tends to 0. As a technical tool to see this, vector-valued Dirichlet series (that is, series with coefficients in some Banach space) are introduced, and some basic definitions and properties (such as the convergence abscissas, Bohr-Cahen formulas) are given.
This chapter looks at how seismic wave theory relates to transforming seismic wave travel-time data into different representations such as the frequency domain (achieved with a 1D Fourier transform), the frequency-wavenumber domain (achieved with a 2D Fourier transform), and the tau-p domain (or intercept time–ray parameter domain). The reason for transforming seismic data into different domains is that the data may be easier to analyze and interpret in other domains. Furthermore, 1D and 2D filtering can be done often more conveniently in the frequency and frequency-wavenumber domains. Also covered are topics related to the tau-p domain, namely, slant-stacking, plane wave decomposition, and the Hilbert and Radon transforms.
We use a variant of a technique used by M. T. Lacey to give sparse
$L^{p}(\log (L))^{4}$
bounds for a class of model singular and maximal Radon transforms.
We study the classical Rosenthal–Szasz inequality for a plane whose geometry is determined by a norm. This inequality states that the bodies of constant width have the largest perimeter among all planar convex bodies of given diameter. In the case where the unit circle of the norm is given by a Radon curve, we obtain an inequality which is completely analogous to the Euclidean case. For arbitrary norms we obtain an upper bound for the perimeter calculated in the anti-norm, yielding an analogous characterisation of all curves of constant width. To derive these results, we use methods from the differential geometry of curves in normed planes.
We study inversion of the spherical Radon transform with centres on a sphere (the data acquisition set). Such inversions are essential in various image reconstruction problems arising in medical, radar and sonar imaging. In the case of radially incomplete data, we show that the spherical Radon transform can be uniquely inverted recovering the image function in spherical shells. Our result is valid when the support of the image function is inside the data acquisition sphere, outside that sphere, as well as on both sides of the sphere. Furthermore, in addition to the uniqueness result, our method of proof provides reconstruction formulas for all those cases. We present a robust computational algorithm and demonstrate its accuracy and efficiency on several numerical examples.
Acquiring conventional 3 km towed streamer data along a 2D profile in the North of Shetland (UK) enables us to use the local Radon-attributes within the context of depth processing methodology for accurate delineation of volcanic units and imaging beneath high-velocity layers. The objective is to map the radially-dipping structure of the Erlend pluton and to investigate the potential existence of relatively soft Cretaceous sediments underneath volcanic units. Success in the Erlend Volcano study requires strict attention to the separation between different groups of events. The crucial point is the generalized discrete Radon transform formulated in terms of local wavefront (dip and curvature) characteristics. This transform is utilized during pre-CMP processing and migration to minimize event-coupling artefacts. These artefacts represent cross-talk energy between various wave modes and include the unwanted part of the wavefield. We show how to produce detailed subsurface images within the region of interest (exploration prospect only) by applying the closely tied processes of prestack event enhancement and separation, well-driven time processing for velocity model building, and final event-based prestack depth imaging. Results show enhanced structural detail and good continuity of principal volcanic units and deeper reflections, suggesting a faulted 0.6 – 0.9 km thick layer of Cretaceous sediments in the proximity of well 209/09-1. Our interpretation complements existing low-resolution geophysical models inferred from gravity and wide-angle seismic data alone.
Johnson and Schechtman (2009) characterized superreflexivity in terms of finite diamond graphs. The present author characterized the Radon–Nikodým property (RNP) for dual spaces in terms of the infinite diamond. This paper is devoted to further study of relations between metric characterizations of superreflexivity and the RNP for dual spaces. The main result is that finite subsets of any set
$M$
whose embeddability characterizes the RNP for dual spaces, characterize superreflexivity. It is also observed that the converse statement does not hold and that
$M\,=\,{{\ell }_{2}}$
is a counterexample.
We make some comments on the existence, uniqueness and integrability of the scalar derivatives and approximate scalar derivatives of vector-valued functions. We are particularly interested in the connection between scalar differentiation and the weak Radon–Nikodým property.
Let
${{F}_{2n,2}}$
be the free nilpotent Lie group of step two on
$2n$
generators, and let
$\mathbf{P}$
denote the affine automorphism group of
${{F}_{2n,2}}$
. In this article the theory of continuous wavelet transform on
${{F}_{2n,2}}$
associated with
$\mathbf{P}$
is developed, and then a type of radial wavelet is constructed. Secondly, the Radon transform on
${{F}_{2n,2}}$
is studied, and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform; the others are from the group Fourier transform. By using wavelet transforms we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. In particular, if
$n\,=\,1$
,
${{F}_{2,2}}$
is the 3-dimensional Heisenberg group
${{H}^{1}}$
, the inversion formula of the Radon transform is valid, which is associated with the sub-Laplacian on
${{F}_{2,2}}$
. This result cannot be extended to the case
$n\,\ge \,2$
.
Radon (222Rn) is a
radioactive gas that occurs naturally when uranium in soil and rock breaks down. Long-term
exposure to 222Rn
increases the risk of lung cancer. The principal objective of this work was to determine
the 222Rn activity
concentration in an indoor air environment at the El-Dabaa site proposed for a nuclear
power plant project in Egypt using the track etch technique with LR115 detectors. The
annual average indoor 222Rn activity concentration in apartment buildings varies
from 24 to 77 Bq.m-3, with a mean value of 54 Bq.m-3. The annual effective dose
received by residents of the studied area was estimated to be 0.41 mSv. The annual
estimated effective dose is less than the recommended action level by ICRP
(3−10 mSv.y-1). The results from this work
provide a radiological assessment program and update the background of the natural
radioactivity map at the El-Dabaa site.
The incredible variety of galaxy shapes cannot be summarized by human defined discrete classes of shapes without causing a possibly large loss of information. Dictionary learning and sparse coding allow us to reduce the high dimensional space of shapes into a manageable low dimensional continuous vector space. Statistical inference can be done in the reduced space via probability distribution estimation and manifold estimation.
We endow certain GKZ-hypergeometric systems with a natural structure of a mixed Hodge module, which is compatible with the mixed Hodge module structure on the Gauß–Manin system of an associated family of Laurent polynomials. As an application we show that the underlying perverse sheaf of a GKZ-system with rational parameter has quasi-unipotent local monodromy.
We prove that localization operators associated to ridgelet transforms with
Lp symbols are bounded
linear operators on L2(Rn).
Operators closely related to these localization operators are shown to be in the trace
class and a trace formula for them is given.
Beach Sand Mining (BSM) is a profitable industry earning a sizable income for the country
by way of foreign exchange. The Indian coast is rich in rare earths such as ilmenite,
rutile, leucoxene, zircon, garnet and sillimanite, and is invariably associated with
radioactive monazite. Due to the nature of the separation processes involved and the
manual handling, workers in these factories are continuously being exposed to suspended
particles containing naturally occurring radioactive materials. An attempt was made to
estimate DNA damage using a chromosome aberration assay to monitor radiation effects in
workers of BSM industries in India. The study group comprised 27 BSM workers and 20
controls. Percentage yields of dicentrics, acentric fragments and chromatid breaks
observed in the control group were 0.058 ± 0.017, 0.073 ± 0.03 and 0.22 ± 0.112,
respectively. Percentage yields of dicentrics + centric rings, acentric fragments and
chromatid breaks observed in the BSM group were 0.029 ± 0.01 (P value 0.19), 0.24 ± 0.06
(P value 0.006) and 0.455 ± 0.06 (P value 0.0004), respectively. Elevated levels of
fragments and chromatid aberrations are suggestive of low-dose radiation effects and also
chemically-induced DNA damage.
with
${{f}_{1}},...,\,{{f}_{n}}$
being polynomials with integer coefficients in the variables
${{x}_{1}},...,\,{{x}_{k}}$
and
${{y}_{1}},...,\,{{y}_{k}}$
. We prove that
${{\sigma }_{\mathbb{Z}}}\left( k \right)\,\ge \,\Omega \left( {{k}^{{6}/{5}\;}} \right)$
.