Proof of Stenger's conjecture on matrix I(-1) of Sinc methods

In this paper, we prove a conjecture, which was proposed by Frank Stenger in 1997, concerning the localization of eigenvalues of the Sinc matrix I^(^-^1^), a problem that is important in both the theory and the practice of Sinc methods. In 2003, Iyad Abu-Jeib and Thomas Shores established a partial answer to this unsolved problem. The techniques they have developed, however, turn out to be the key that finally leads to the settlement here of Stenger's conjecture.

[1]  Kenneth L. Bowers,et al.  Sinc methods for quadrature and differential equations , 1987 .

[2]  Gerd Baumann,et al.  Fractional calculus and Sinc methods , 2011 .

[3]  Masaaki Sugihara,et al.  Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind , 2010, J. Comput. Appl. Math..

[4]  S. V. Savchenko.  Typical Changes in Spectral Properties under Perturbations by a Rank-One Operator , 2003 .

[5]  Froilán M. Dopico,et al.  Low Rank Perturbation of Jordan Structure , 2003, SIAM J. Matrix Anal. Appl..

[6]  Mehdi Dehghan,et al.  The Sinc-collocation method for solving the Thomas-Fermi equation , 2013, J. Comput. Appl. Math..

[7]  S. V. Savchenko.  On the Change in the Spectral Properties of a Matrix under Perturbations of Sufficiently Low Rank , 2004 .

[8]  William F. Trench,et al.  Some Spectral Properties of Hermitian Toeplitz Matrices , 1994 .

[9]  F. Stenger.  Handbook of Sinc Numerical Methods , 2010 .

[10]  Shmuel Friedland,et al.  Eigenvalues of Almost Skew Symmetric Matrices and Tournament Matrices , 1993 .

[11]  A. Cantoni,et al.  Eigenvalues and eigenvectors of symmetric centrosymmetric matrices , 1976 .

[12]  F. Stenger.  Numerical Methods Based on Sinc and Analytic Functions , 1993 .

[13]  L. Hörmander,et al.  A Remark on Perturbations of Compact Operators. , 1994 .

[14]  Kenneth L. Bowers,et al.  The symmetric Sinc-Galerkin method yields ADI model problems , 2012, Appl. Math. Comput..

[15]  Leiba Rodman,et al.  Eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations , 2011 .

[16]  Rakesh K. Kapania,et al.  Imposing boundary conditions in Sinc method using highest derivative approximation , 2009 .

[17]  Frank Stenger.  Matrices of Sinc methods , 1997 .

[18]  V. N. Bogaevski,et al.  Matrix Perturbation Theory , 1991 .