scholarly journals A New Application Methodology of the Fourier Transform for Rational Approximation of the Complex Error Function

2016 ◽  
Vol 8 (1) ◽  
pp. 14 ◽  
Author(s):  
S. M. Abrarov ◽  
B. M. Quine
Keyword(s):  
Fourier Transform ◽  
Rational Approximation ◽  
Error Function ◽  
Practical Importance ◽  
Exponential Functions ◽  
New Approach ◽  
Practical Applications ◽  
Input Parameters ◽  
The Fourier Transform

<p>This paper presents a new approach in application of the Fourier transform to the complex error function resulting in an efficient rational approximation. Specifically, the computational test shows that with only $17$ summation terms the obtained rational approximation of the complex error function provides accuracy ${10^{ - 15}}$ over the most domain of practical importance $0 \le x \le 40,000$ and ${10^{ - 4}} \le y \le {10^2}$ required for the HITRAN-based spectroscopic applications. Since the rational approximation does not contain trigonometric or exponential functions dependent upon the input parameters $x$ and $y$, it is rapid in computation. Such an example demonstrates that the considered methodology of the Fourier transform may be advantageous in practical applications.</p>

Two notes on spectral synthesis for discrete Abelian groups

1965 ◽  
Vol 61 (3) ◽  
pp. 617-620 ◽  
Author(s):  
R. J. Elliott
Keyword(s):  
Fourier Transform ◽  
Dual Space ◽  
Fourier Transforms ◽  
Spectral Synthesis ◽  
Division Problem ◽  
Exponential Functions ◽  
Translation Invariant ◽  
Annihilator Ideal ◽  
The Fourier Transform ◽  
Translation Invariant Subspace

Introduction. Spectral synthesis is the study of whether functions in a certain set, usually a translation invariant subspace (a variety), can be synthesized from certain simple functions, exponential monomials, which are contained in the set. This problem is transformed by considering the annihilator ideal in the dual space, and after taking the Fourier transform the problem becomes one of deciding whether a function is in a certain ideal, that is, we have a ‘division problem’. Because of this we must take into consideration the possibility of the Fourier transforms of functions having zeros of order greater than or equal to 1. This is why, in the original situation, we study whether varieties are generated by their exponential monomials, rather than just their exponential functions. This viewpoint of the problem as a division question, of course, perhaps throws light on why Wiener's Tauberian theorem works, and is implicit in the construction of Schwartz's and Malliavin's counter examples to spectral synthesis in L1(G) (cf. Rudin ((4))).

scholarly journals IDENTIFICATION OF BINARY CELLULAR AUTOMATA FROM SPATIO-TEMPORAL BINARY PATTERNS USING A FOURIER REPRESENTATION

2007 ◽  
Vol 17 (06) ◽  
pp. 1985-1996 ◽  
Author(s):  
L. Z. GUO ◽  
S. A. BILLINGS
Keyword(s):  
Fourier Transform ◽  
Cellular Automata ◽  
Least Squares ◽  
Boolean Functions ◽  
Multilinear Polynomial ◽  
Polynomial Representation ◽  
New Approach ◽  
Spatio Temporal ◽  
The Fourier Transform ◽  
Robust To Noise

The identification of binary cellular automata from spatio-temporal binary patterns is investigated in this paper. Instead of using the usual Boolean or multilinear polynomial representation, the Fourier transform representation of Boolean functions is employed in terms of a Fourier basis. In this way, the orthogonal forward regression least-squares algorithm can be applied directly to detect the significant terms and to estimate the associated parameters. Compared with conventional methods, the new approach is much more robust to noise. Examples are provided to illustrate the effectiveness of the proposed approach.

A NANOSTRUCTURED POLYMORPH OF μ-OXOBIS(PHTHALOCYANINATOIRON(III)) STUDIED BY ANGULAR AND ENERGY DISPERSIVE X-RAY DIFFRACTION

NANO ◽  
2007 ◽  
Vol 02 (02) ◽  
pp. 121-128 ◽  
Author(s):  
ROBERTO MATASSA ◽  
PAOLO BALLIRANO ◽  
MARIA PIA DONZELLO ◽  
CLAUDIO ERCOLANI ◽  
CLAUDIA SADUN ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Rietveld Method ◽  
Energy Dispersive ◽  
Structural Function ◽  
X Ray Diffraction ◽  
New Approach ◽  
X Ray ◽  
The Fourier Transform

A new approach of X-ray diffraction was used to investigate the nanostructured μ-Oxo(2) polymorph of μ-oxo-bis(phthalocyaninatoiron(III)), [ PcFe – O – FePc ]. The packing of the dinuclear units was determined by the Rietveld method on Angular Dispersive X-ray Diffraction (ADXD) data, whereas the intramolecular geometry was optimized by Energy Dispersive X-ray Diffraction (EDXD) exploiting the peculiar strength of those techniques. The dimension of the nanoparticles was estimated from the Fourier transform of the EDXD experimental structural function.

Analysis of a Magnetic Sphere Using the Fourier Transform: A New Approach

1982 ◽  
Vol 13 (2) ◽  
pp. 35-40 ◽  
Author(s):  
N. L. Mohan ◽  
N. B. R. Prasad ◽  
S. V. Seshagari Rao ◽  
V. L. S. Bhimasankaram
Keyword(s):  
Fourier Transform ◽  
New Approach ◽  
The Fourier Transform ◽  
Magnetic Sphere

scholarly journals A Rational Approximation of the Fourier Transform by Integration with Exponential Decay Multiplier

2021 ◽  
Vol 12 (11) ◽  
pp. 947-962
Author(s):  
Sanjar M. Abrarov ◽  
Rehan Siddiqui ◽  
Rajinder K. Jagpal ◽  
Brendan M. Quine
Keyword(s):  
Fourier Transform ◽  
Rational Approximation ◽  
Exponential Decay ◽  
The Fourier Transform

scholarly journals Time reversibility in acoustic signals

2004 ◽  
Vol 2 (02) ◽  
Author(s):  
J.M. Velázquez ◽  
A. Ramí­rez ◽  
C. A. Vargas
Keyword(s):  
Fourier Transform ◽  
Green’S Function ◽  
Green's Function ◽  
Acoustic Signals ◽  
Sound Waves ◽  
Time Reversibility ◽  
Practical Applications ◽  
The Fourier Transform

In this work we make a description of the time reversion problem in sound waves. Our objective is to explain the phenomena through the Fourier transform of the Green's function. With this function it is possible to characterize the propagation of the emitted signal. It also can be used to express the time-reversed signal in such a way that we can select precisely the destination site of the signal. Finally, we show some possible practical applications of this problem.

scholarly journals One Technique to Enhance the Resolution of Discrete Fourier Transform

Electronics ◽  
2019 ◽  
Vol 8 (3) ◽  
pp. 330 ◽  
Author(s):  
Ivan Kanatov ◽  
Dmitry Kaplun ◽  
Denis Butusov ◽  
Viacheslav Gulvanskii ◽  
Aleksander Sinitca
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Digital Signal ◽  
Analysis Tool ◽  
Signal Phase ◽  
Practical Applications ◽  
Advantages And Disadvantages ◽  
Engine Vibration ◽  
Window Functions ◽  
The Fourier Transform

Discrete Fourier transform (DFT) is a common analysis tool in digital signal processing. This transform is well studied and its shortcomings are known as well. Various window functions (e.g., Hanning, Blackman, Kaiser) are often used to reduce sidelobes and to spread the spectrum. In this paper, we introduce a transformation that allows removing the sidelobes of the Fourier transform and increasing the resolution of the DFT without changing the time sample. The proposed method is based on signal phase analysis. We give the comparison of the proposed approach with known methods based on window functions. The advantages and disadvantages of the proposed technique are explicitly shown. We also give a set of examples illustrating the application of our technique in some practical applications, including engine vibration analysis and a short-range radar system.

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