What Is It Possible to Say About an Asymptotic of the Fourier Transform of the Characteristic Function of a Two-dimensional Convex Body with Nonsmooth Boundary?

2004 ◽  
pp. 209-215 ◽  
Author(s):  
A. N. Podkorytov
Keyword(s):  
Fourier Transform ◽  
Convex Body ◽  
Characteristic Function ◽  
Two Dimensional ◽  
Nonsmooth Boundary ◽  
The Fourier Transform

scholarly journals Two-Dimensional Quaternion Linear Canonical Transform: Properties, Convolution, Correlation, and Uncertainty Principle

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Mawardi Bahri ◽  
Ryuichi Ashino
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Linear Canonical Transform ◽  
Two Dimensional ◽  
Quaternion Fourier Transform ◽  
Gaussian Functions ◽  
The Fourier Transform ◽  
Definition Of ◽  
Quaternion Linear Canonical Transform

A definition of the two-dimensional quaternion linear canonical transform (QLCT) is proposed. The transform is constructed by substituting the Fourier transform kernel with the quaternion Fourier transform (QFT) kernel in the definition of the classical linear canonical transform (LCT). Several useful properties of the QLCT are obtained from the properties of the QLCT kernel. Based on the convolutions and correlations of the LCT and QFT, convolution and correlation theorems associated with the QLCT are studied. An uncertainty principle for the QLCT is established. It is shown that the localization of a quaternion-valued function and the localization of the QLCT are inversely proportional and that only modulated and shifted two-dimensional Gaussian functions minimize the uncertainty.

On: “Gravity Interpretation Using the Fourier Integral”, by M. E. Odegard and J. W. Berg, Jr. (GEOPHYSICS, June 1965, p. 127–138)

Geophysics ◽  
1975 ◽  
Vol 40 (2) ◽  
pp. 356-357
Author(s):  
Jay Gopal Saha
Keyword(s):  
Fourier Transform ◽  
Gravity Anomaly ◽  
Fourier Integral ◽  
Two Dimensional ◽  
Transform Method ◽  
Reverse Problem ◽  
Vertical Fault ◽  
Vertical Step ◽  
The Fourier Transform ◽  
Gravity Interpretation

In their paper, Odegard and Berg claim that from the gravity anomaly due to a two‐dimensional vertical fault the density, the throw, and the depth can be determined uniquely by a Fourier transform method. It is true that the solution of the reverse problem for a two‐dimensional vertical step is theoretically unique. The derivation of the Fourier transform by the authors, however, is erroneous.

The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters

1996 ◽  
Vol 16 (2) ◽  
pp. 207-253 ◽  
Author(s):  
Mustafa Akcoglu ◽  
Alexandra Bellow ◽  
Roger L. Jones ◽  
Viktor Losert ◽  
Karin Reinhold-Larsson ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Characteristic Function ◽  
Ergodic Averages ◽  
Riemann Sums ◽  
Lacunary Sequences ◽  
Linearly Independent ◽  
Convolution Powers ◽  
The Fourier Transform

AbstractIn this paper we establish conditions on a sequence of operators which imply divergence. In fact, we give conditions which imply that we can find a set B of measure as close to zero as we like, but such that the operators applied to the characteristic function of this set have a lim sup equal to 1 and a lim inf equal to 0 a.e. (strong sweeping out). The results include the fact that ergodic averages along lacunary sequences, certain convolution powers, and the Riemann sums considered by Rudin are all strong sweeping out. One of the criteria for strong sweeping out involves a condition on the Fourier transform of the sequence of measures, which is often easily checked. The second criterion for strong sweeping out involves showing that a sequence of numbers satisfies a property similar to the conclusion of Kronecker's lemma on sequences linearly independent over the rationals.

Possible Models of Murein and Their Fourier Transforms

1982 ◽  
Vol 37 (3-4) ◽  
pp. 226-235 ◽  
Author(s):  
Helmut Formanek
Keyword(s):  
Fourier Transform ◽  
Cell Walls ◽  
Fourier Transforms ◽  
Elementary Cell ◽  
Two Dimensional ◽  
Density Measurements ◽  
X Ray ◽  
Rigid Layer ◽  
The Fourier Transform ◽  
Almost All

Abstract Murein, Models, Fourier Transforms Murein, the rigid layer of the cell walls of almost all bacteria can be regarded as derivative of chitin. Within the sterically allowed region its polysaccharide chain can perform conformations with two-to threefold screw axes. Two dimensional Fourier transforms calculated from different possible conformations have been compared with data of density measurements, X-ray and electron diffraction. The Fourier transform of murein with a chitin-like conformation of the poly­ saccharide chain and an elementary cell of 4.5 × 10.4 × 21.5 Å3 provides the best agreement with the experimental results.

Introduction

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Nuclear Magnetic Resonance ◽  
Fourier Transform ◽  
Magnetic Resonance ◽  
Fourier Analysis ◽  
Fourier Integral ◽  
Sampling Theorem ◽  
Two Dimensional ◽  
The Fourier Transform ◽  
Array Imaging ◽  
Nuclear Magnetic

Jean Baptiste Joseph Fourier’s powerful idea of decomposition of a signal into sinusoidal components has found application in almost every engineering and science field. An incomplete list includes acoustics [1497], array imaging [1304], audio [1290], biology [826], biomedical engineering [1109], chemistry [438, 925], chromatography [1481], communications engineering [968], control theory [764], crystallography [316, 498, 499, 716], electromagnetics [250], imaging [151], image processing [1239] including segmentation [1448], nuclear magnetic resonance (NMR) [436, 1009], optics [492, 514, 517, 1344], polymer characterization [647], physics [262], radar [154, 1510], remote sensing [84], signal processing [41, 154], structural analysis [384], spectroscopy [84, 267, 724, 1220, 1293, 1481, 1496], time series [124], velocity measurement [1448], tomography [93, 1241, 1242, 1327, 1330, 1325, 1331], weather analysis [456], and X-ray diffraction [1378], Jean Baptiste Joseph Fourier’s last name has become an adjective in the terms like Fourier series [395], Fourier transform [41, 51, 149, 154, 160, 437, 447, 926, 968, 1009, 1496], Fourier analysis [151, 379, 606, 796, 1472, 1591], Fourier theory [1485], the Fourier integral [395, 187, 1399], Fourier inversion [1325], Fourier descriptors [826], Fourier coefficients [134], Fourier spectra [624, 625] Fourier reconstruction [1330], Fourier spectrometry [84, 355], Fourier spectroscopy [1220, 1293, 1438], Fourier array imaging [1304], Fourier transform nuclear magnetic resonance (NMR) [429, 1004], Fourier vision [1448], Fourier optics [419, 517, 1343], and Fourier acoustics [1496]. Applied Fourier analysis is ubiquitous simply because of the utility of its descriptive power. It is second only to the differential equation in the modelling of physical phenomena. In contrast with other linear transforms, the Fourier transform has a number of physical manifestations. Here is a short list of everyday occurrences as seen through the lens of the Fourier paradigm. • Diffracting coherent waves in sonar and optics in the far field are given by the two dimensional Fourier transform of the diffracting aperture. Remarkably, in free space, the physics of spreading light naturally forms a two dimensional Fourier transform. • The sampling theorem, born of Fourier analysis, tells us how fast to sample an audio waveform to make a discrete time CD or an image to make a DVD.

STATISTICAL BEHAVIOR OF CHARACTERISTIC LENGTHS OF VIBRATIONS ON TWO-DIMENSIONAL RANDOM FRACTALS

Fractals ◽  
1993 ◽  
Vol 01 (04) ◽  
pp. 1044-1050 ◽  
Author(s):  
ALBERTO PETRI ◽  
GIANCARLO RUOCCO
Keyword(s):  
Fourier Transform ◽  
Generalized Inverse ◽  
Scaling Law ◽  
Normal Modes ◽  
Two Dimensional ◽  
Maximum Weight ◽  
Statistical Behavior ◽  
Apparent Correlation ◽  
Expected Values ◽  
The Fourier Transform

We analyze the behavior of some characteristic lengths of the normal modes of square percolating lattices at threshold. Generalized inverse participation ratios are taken as estimates of the localization lengths, while the wavenumber kM having the maximum weight in the Fourier transform of the autocorrelation function of each mode is supposed to be inversely proportional to the relevant wavelength. Both site and bond percolating lattices are considered. It is found that, on average, these quantities follow about the same supposed scaling law, therefore supporting the idea of a universal length. At the same time they all exhibit large deviations from the expected values with no apparent correlation among fluctuations in the localization lengths and in the relevant wavelengths, as it turns out from the computed correlation coefficient.

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