scholarly journals On the Invalidity of Fourier Series Expansions of Fractional Order

2015 ◽  
Vol 18 (6) ◽  
Author(s):  
Peter R. Massopust ◽  
Ahmed I. Zayed
Keyword(s):  
Fourier Series ◽  
Periodic Function ◽  
Series Expansion ◽  
Fractional Order ◽  
Fourier Series Expansion ◽  
Short Paper ◽  
Series Expansions ◽  
Exponential Functions ◽  
Smooth Periodic Function

AbstractThe purpose of this short paper is to show the invalidity of a Fourier series expansion of fractional order as derived by G. Jumarie in a series of papers. In his work the exponential functions eHe showed that any smooth periodic function f with period M

Fractional-order Fourier series expansion for the analysis of chirped pulses

2005 ◽  
Vol 249 (1-3) ◽  
pp. 145-152 ◽  
Author(s):  
Sébastien Coëtmellec ◽  
Marc Brunel ◽  
Denis Lebrun ◽  
Jean-Bernard Lecourt
Keyword(s):  
Fourier Series ◽  
Series Expansion ◽  
Fractional Order ◽  
Fourier Series Expansion ◽  
Chirped Pulses

ANALYSIS OF SPIDER INTERFEROGRAMS WITH THE FRACTIONAL-ORDER FOURIER SERIES EXPANSION

2006 ◽  
Vol 15 (04) ◽  
pp. 501-511
Author(s):  
MARC BRUNEL ◽  
SEBASTIEN COETMELLEC
Keyword(s):  
Fourier Series ◽  
Series Expansion ◽  
Fractional Order ◽  
Ultrashort Pulses ◽  
Fourier Series Expansion ◽  
Third Order ◽  
Third Order Dispersion ◽  
Order Dispersion

We propose the use of fractional-order Fourier series expansion for the characterization of ultrashort pulses. The analysis and reconstruction of simulated SPIDER interferograms is demonstrated with this decomposition process. We consider the case of ultrashort pulses (about 100 fs) that have been subject to second- and third-order dispersion. Our technique allows the simultaneous determination of the quadratic and cubic spectral phase coefficients of the pulses. The technique can be further used to analyze, filter or reconstruct noisy SPIDER interferograms.

A SIMPLE MODEL FOR ORIENTATIONAL ORDERING OF A SILVER MONOLAYER ELECTRODEPOSITED ON A C(0001) SURFACE

1995 ◽  
Vol 02 (04) ◽  
pp. 489-494 ◽  
Author(s):  
E.E. MOLA ◽  
A.G. APPIGNANESSI ◽  
J.L. VICENTE ◽  
L. VAZQUEZ ◽  
R.C. SALVAREZZA ◽  
...  
Keyword(s):  
Experimental Data ◽  
Fourier Series ◽  
Simple Model ◽  
Series Expansion ◽  
Unit Cell ◽  
Fourier Series Expansion ◽  
Crystal Formation ◽  
Substrate Potential ◽  
Weber Type ◽  
Orientational Ordering

The model for the angular orientational energy (AOE) has been extended to hexagonal submonolayer domains of Ag electrodeposited at a constant overpotential on a C(0001) surface. These domains which are characterized by an epitaxy angle θ=15±5° and an Ag−Ag distance d Ag−Ag =0.330± 0.016 nm, can be considered as precursors of 3D Ag crystal formation, according to a Volmer-Weber type mechanism. Calculations are based upon a simple Hamiltonian evaluated by introducing the concept of the commensurable unit cell. A Fourier series expansion for the substrate potential was used. Results from the model predict the existence of a commensurable cell in agreement with the experimental data derived from STM imaging.

Predicting Apple Firmness and Soluble Solids Content Based on Hyperspectral Scattering Imaging Using Fourier Series Expansion

2017 ◽  
Vol 60 (4) ◽  
pp. 1053-1062
Author(s):  
Wei Wang ◽  
Min Huang ◽  
Qibing Zhu
Keyword(s):  
Fourier Series ◽  
Least Squares ◽  
Series Expansion ◽  
Fourier Coefficients ◽  
Expansion Method ◽  
Soluble Solids ◽  
Fourier Series Expansion ◽  
Support Vector ◽  
Soluble Solids Content ◽  
Solids Content

Abstract. This article reports on using a Fourier series expansion method to extract features from hyperspectral scattering profiles for apple fruit firmness and soluble solids content (SSC) prediction. Hyperspectral scattering images of ‘Golden Delicious’ (GD), ‘Jonagold’ (JG), and ‘Delicious’ (RD) apples, harvested in 2009 and 2010, were acquired using an online hyperspectral imaging system over the wavelength region of 500 to 1000 nm. The moment method and Fourier series expansion method were used to analyze the scattering profiles of apples. The zeroth-first order moment (Z-FOM) spectra and Fourier coefficients were extracted from each apple, which were then used for developing fruit firmness and SSC prediction models using partial least squares (PLS) and least squares support vector machine (LSSVM). The PLS models based on the Fourier coefficients improved the standard errors of prediction (SEP) by 4.8% to 19.9% for firmness and by 2.4% to 13.5% for SSC, compared with the PLS models using the Z-FOM spectra. The LSSVM models for the prediction set of Fourier coefficients achieved better SEP results, with improvements of 4.4% to 11.3% for firmness and 2.8% to 16.5% for SSC over the LSSVM models for the Z-FOM spectra data and 3.7% to 12.6% for firmness and 5.4% to 8.6% for SSC over the PLS models for the Fourier coefficients. Experiments showed that Fourier series expansion provides a simple, fast, and effective means for improving Keywords: Apples, Firmness, Fourier series expansion, Hyperspectral scattering imaging, Least squares support vector machine, Partial least squares, Soluble solids content.

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