IEICE TRANSACTIONS on Fundamentals
Resolution of the Gibbs Phenomenon for Fractional Fourier Series
Summary :
The nth partial sums of a classical Fourier series have large oscillations near the jump discontinuities. This behaviour is the well-known Gibbs phenomenon. Recently, the inverse polynomial reconstruction method (IPRM) has been successfully implemented to reconstruct piecewise smooth functions by reducing the effects of the Gibbs phenomenon for Fourier series. This paper addresses the 2-D fractional Fourier series (FrFS) using the same approach used with the 1-D fractional Fourier series and finds that the Gibbs phenomenon will be observed in 1-D and 2-D fractional Fourier series expansions for functions at a jump discontinuity. The existing IPRM for resolution of the Gibbs phenomenon for 1-D and 2-D FrFS appears to be the same as that used for Fourier series. The proof of convergence provides theoretical basis for both 1-D and 2-D IPRM to remove Gibbs phenomenon. Several numerical examples are investigated. The results indicate that the IPRM method completely eliminates the Gibbs phenomenon and gives exact reconstruction results.
- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E97-A No.2 pp.572-586
- Publication Date
- 2014/02/01
- Publicized
- Online ISSN
- 1745-1337
- DOI
- 10.1587/transfun.E97.A.572
- Type of Manuscript
- PAPER
- Category
- Digital Signal Processing
Authors
Hongqing ZHU
East China University of Science and Technology
Meiyu DING
East China University of Science and Technology
Daqi GAO
East China University of Science and Technology
Keyword
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Hongqing ZHU, Meiyu DING, Daqi GAO, "Resolution of the Gibbs Phenomenon for Fractional Fourier Series" in IEICE TRANSACTIONS on Fundamentals,
vol. E97-A, no. 2, pp. 572-586, February 2014, doi: 10.1587/transfun.E97.A.572.
Abstract: The nth partial sums of a classical Fourier series have large oscillations near the jump discontinuities. This behaviour is the well-known Gibbs phenomenon. Recently, the inverse polynomial reconstruction method (IPRM) has been successfully implemented to reconstruct piecewise smooth functions by reducing the effects of the Gibbs phenomenon for Fourier series. This paper addresses the 2-D fractional Fourier series (FrFS) using the same approach used with the 1-D fractional Fourier series and finds that the Gibbs phenomenon will be observed in 1-D and 2-D fractional Fourier series expansions for functions at a jump discontinuity. The existing IPRM for resolution of the Gibbs phenomenon for 1-D and 2-D FrFS appears to be the same as that used for Fourier series. The proof of convergence provides theoretical basis for both 1-D and 2-D IPRM to remove Gibbs phenomenon. Several numerical examples are investigated. The results indicate that the IPRM method completely eliminates the Gibbs phenomenon and gives exact reconstruction results.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E97.A.572/_p
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@ARTICLE{e97-a_2_572,
author={Hongqing ZHU, Meiyu DING, Daqi GAO, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Resolution of the Gibbs Phenomenon for Fractional Fourier Series},
year={2014},
volume={E97-A},
number={2},
pages={572-586},
abstract={The nth partial sums of a classical Fourier series have large oscillations near the jump discontinuities. This behaviour is the well-known Gibbs phenomenon. Recently, the inverse polynomial reconstruction method (IPRM) has been successfully implemented to reconstruct piecewise smooth functions by reducing the effects of the Gibbs phenomenon for Fourier series. This paper addresses the 2-D fractional Fourier series (FrFS) using the same approach used with the 1-D fractional Fourier series and finds that the Gibbs phenomenon will be observed in 1-D and 2-D fractional Fourier series expansions for functions at a jump discontinuity. The existing IPRM for resolution of the Gibbs phenomenon for 1-D and 2-D FrFS appears to be the same as that used for Fourier series. The proof of convergence provides theoretical basis for both 1-D and 2-D IPRM to remove Gibbs phenomenon. Several numerical examples are investigated. The results indicate that the IPRM method completely eliminates the Gibbs phenomenon and gives exact reconstruction results.},
keywords={},
doi={10.1587/transfun.E97.A.572},
ISSN={1745-1337},
month={February},}
Copy
TY - JOUR
TI - Resolution of the Gibbs Phenomenon for Fractional Fourier Series
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 572
EP - 586
AU - Hongqing ZHU
AU - Meiyu DING
AU - Daqi GAO
PY - 2014
DO - 10.1587/transfun.E97.A.572
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E97-A
IS - 2
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - February 2014
AB - The nth partial sums of a classical Fourier series have large oscillations near the jump discontinuities. This behaviour is the well-known Gibbs phenomenon. Recently, the inverse polynomial reconstruction method (IPRM) has been successfully implemented to reconstruct piecewise smooth functions by reducing the effects of the Gibbs phenomenon for Fourier series. This paper addresses the 2-D fractional Fourier series (FrFS) using the same approach used with the 1-D fractional Fourier series and finds that the Gibbs phenomenon will be observed in 1-D and 2-D fractional Fourier series expansions for functions at a jump discontinuity. The existing IPRM for resolution of the Gibbs phenomenon for 1-D and 2-D FrFS appears to be the same as that used for Fourier series. The proof of convergence provides theoretical basis for both 1-D and 2-D IPRM to remove Gibbs phenomenon. Several numerical examples are investigated. The results indicate that the IPRM method completely eliminates the Gibbs phenomenon and gives exact reconstruction results.
ER -
English
