A new strategy for choosing the Chebyshev‐gegenbauer parameters in a reconstruction based on asymptotic analysis

References

• Abramowitz , M. and Stegun , I.S. 1972 . Handbook of mathematical functions with formulas, graphs, and mathematical tables. , Washington : National Bureau of Standards .
   Google Scholar
• Archibald , R. , Chen , K. , Gelb , A. and Renaut , R. 2003 . Improving tissue segmentation of human brain MRI through pre‐processing by the Gegenbauer reconstruction method . NeuroImage, , 20 (1) : 489 – 502 . (doi:10.1016/S1053–8119(03)00260‐X)
   PubMed Web of Science ®Google Scholar
• Archibald , R. and Gelb , A. 2002 . A method to reduce the Gibbs ringing artifact in MRI scans while keeping tissue boundary integrity . IEEE Trans. Med. Imaging, , 21 : 305 – 319 . (doi:10.1109/TMI.2002.1000255)
   PubMed Web of Science ®Google Scholar
• Boyd , J. 2004 . Trouble with Gegenbauer reconstruction for defeating Gibbs’ phenomenon: Runge phenomenon in the diagonal limit of Gegenbauer polynomial approximations . J. Comput. Phys., , 20 : 433 – 459 .
   Google Scholar
• Canuto , C. , Hussaini , M.Y. , Quarteroni , A. and Zang , T.A. 1988 . Spectral Methods in Fluid Dynamics. , New York : Springer‐Verlag .
   Google Scholar
• Cates , D. 2007 . Edge detection using Fourier data with applications. , Arizona State University . Ph.D. Dissertation
   Google Scholar
• Davis , P.J. and Rabinowitz , P. 1984 . Methods of numerical integration. , Ontario : Academic Press .
   Google Scholar
• Driscoll , T. and Fornberg , B. 2001 . A pade‐based algorithm for overcoming the Gibbs phenomenon . Numer. Algorithms, , 26 : 77 – 92 . (doi:10.1023/A:1016648530648)
   Web of Science ®Google Scholar
• Gelb , A. 2004 . On the reduction of round‐off error for the Gegenbauer reconstruction method . J. Sci. Comput., , 20 : 433 – 459 . (doi:10.1023/B:JOMP.0000025933.39334.17)
   Web of Science ®Google Scholar
• Gelb , A. and Jackiewicz , Z. 2005 . Determining analiticity for parameter optimization of the Gegenbauer reconstruction method . SIAM J. Sci. Comput., , 27 : 1014 – 1031 . (doi:10.1137/040603814)
   Web of Science ®Google Scholar
• Gelb , A. and Tanner , J. 2006 . Robust reprojection methods for the resolution of the Gibbs’ phenomenon . Appl. Comput. Harmon. Anal., , 20 : 3 – 25 .
   Web of Science ®Google Scholar
• Gottlieb , D. and Orszag , S.A. 1977 . Numerical analysis of spectral methods: theory and applications. , Philadelphia : SIAM‐CBMS .
   Google Scholar
• Gottlieb , D. and Shu , C.W. 1995 . On the Gibbs phenomenon IV: recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic function . Math. Comput., , 64 : 1081 – 1095 . (doi:10.2307/2153484)
   Web of Science ®Google Scholar
• Gottlieb , D. and Shu , C.W. 1995 . On the Gibbs phenomenon V: recovering exponential accuracy from collocation point values of a piecewise analytic function . Numer. Math., , 71 : 511 – 526 . (doi:10.1007/s002110050155)
   Web of Science ®Google Scholar
• Gottlieb , D. and Shu , C.W. 1997 . On the Gibbs phenomenon and its resolution . SIAM Rev., , 39 : 644 – 668 . (doi:10.1137/S0036144596301390)
   Web of Science ®Google Scholar
• Gottlieb , D. and Shu , C.W. 1998 . A general theory for the resolution of the Gibbs phenomenon . Atti dei Convegni Lincei, , 147 : 39 – 48 .
   Google Scholar
• Gottlieb , D. , Shu , C.W. , Solomonoff , A. and Vandeven , H. 1992 . On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function . J. Comput. Appl. Math., , 43 : 81 – 98 . (doi:10.1016/0377–0427(92)90260–5)
   Google Scholar
• Hesthaven , J.S. , Gottlieb , S. and Gottlieb , D. 2002 . Spectral methods for time dependent problems. , Providence, Rhode Island : Cambridge University Press .
   Google Scholar
• Jackiewicz , Z. 2003 . Determination of optimal parameters for the Chebyshev‐Gegenbauer reconstruction method . SIAM J. Sci. Comput., , 25 : 1187 – 1198 . (doi:.1137/S1064827503423597)
   Web of Science ®Google Scholar
• Jackiewicz , Z. and Park , R. 2009 . A strategy for choosing Gegenbauer reconstruction parameters for numerical stability . Appl. Math. Comput., , 212 : 418 – 434 . (doi:10.1016/j.amc.2009.02.034)
   Web of Science ®Google Scholar
• Park , R. 2009 . Optimal compression and numerical stability for Gegenbauer reconstructions with applications. , Arizona State University . Ph.D. Dissertation
   Google Scholar
• Shizgal , B. and Jung , J.H. 2003 . Towards the resolution of the Gibbs phenomena . J. Comput. Appl. Math., , 161 : 41 – 65 . (doi:10.1016/S0377–0427(03)00500–4)
   Web of Science ®Google Scholar
• Vandeven , H. 1991 . Family of spectral filters for discontinuous problems . J. Sci. Comput., , 6 : 159 – 192 . (doi:10.1007/BF01062118)
   Google Scholar