Abstract
Banded lower triangular ℕ×ℕ Toeplitz matrices A are considered. A sufficient condition for the elements of A−1 to decay to 0 fast is given. Moreover, some bounds of the norms of these inverses are also found.
Similar content being viewed by others
References
Ch. Affane-Aji, N. Agarwal and N. V. Govil, Location of zeros of polynomials, Math. Comput. Model, 50 (2009), 306–313.
K. S. Berenhaut, D. C. Morton and P. T. Fletcher, Bounds for inverses of triangular Toeplitz matrices, SIAM J. Matrix Anal. Appl., 27 (2005), 212–217.
A. Böttcher and S. M. Grudsky, Toeplitz matrices, asymptotic linear algebra and functional analysis, Texts and Readings in Mathematics 18, Hindustan Book Agency, 2000.
S. Demko, Inverses of band matrices and local convergence of spline projections, SIAM J. Numer. Anal., 14 (1977), 616–619.
S. Demko, W. F. Moss and P. W. Smith, Decay rates for inverses of band matrices, Math. Comput., 43 (1984), 491–499.
G. Enestrôm, Remark on a theorem on the roots of the equation anxn + an-1xn-1 + • • • + a1x + a0 = 0 where all coefficients are real and positive, Tôhoku Math. J., 18 (1920), 34–36.
N. J. Ford, D. V. Savostyanov and N. I. Zamarshkin, On the decay of the elements of inverse triangular Topelitz matrices, SIAM J. Matrix Anal. Appl., 35 (2014), 1288–1302.
R. B. Gardner and N. K. Govil, Eneström-Kakeya theorem and some of its generalizations, Current topics in pure and computational complex analysis (Trends Math.), Birkhäuser/Springer, New Delhi, 2014.
S. Grudsky and A. Rybkin, Soliton theory and Hankel operators, SIAM J. Math. Anal., 47 (2015), 2283–2323.
G. H. Hardy, Divergent series, Clarendon Press, Oxford, UK, 1949.
S. Jaffard, Propriétés des matrices “bien localisées” prés de leur diagonale et quelques applications, Ann. Inst. H. Poincaré Anal. Non Linéire, 7 (1990), 461476.
S. Kakeya, On the limits of the roots of an algebraic equation with positive coefficients, Tôhoku Math. J., 2 (1912–3), 140–142.
D. Kershew, Inequalities on the elements of the inverse of a certain tridiagonal matrix, Math. Comp., 24 (1970), 155–158.
S.- H. Kim, On the moduli of the zeros of a polynomial, Amer. Math. Monthly, 112 (2005), 924–925.
X. Liu, S. McKee, J.Y. Yuan and X. Y. Yuan, Uniform bounds on the 1-norm of the inverse of lower triangular Toeplitz matrices, Linear Algebra Appl., 435 (2011), 1157–1170.
T. Lungenstrass and G. Raikov, Local spectral asymptotics for metric perturbations of the Landau Hamiltonian, Anal. PDE, 8 (2015), 1237–1262.
B. Mityagin, Quadratic pencils and least-squares piecewise-polynomial approximation, Math. Comp., 40 (1983), 283–300.
P. Montel, Sur quelques limites pour les modules des zéros des polynômes, Comment. Math. Helv., 7 (1934–35), 178–200.
H.- K. Pang, H.- H. Qin, H.-W. Sun and T.- T. Ma, Circulant-based approximate inverse preconditioners for a class of fractional diffusion equations, Comput. Math. App., 85 (2021), 18–29.
A. Yu. Shadrin, The L∞-norm of the L2-spline projector is bounded independently of the knot sequence: A proof of de Boor’s conjecture, Acta Math., 187 (2001), 59–137.
D. M. Simeunović, On the location of the zeros of polynomials, Math. Moravica, 2 (1998), 91–96.
A. Vecchio, A bound for the inverse of a lower triangular Topelitz matrix, SIAM J. Matrix Anal. Appl., 24 (2003), 1167–1174.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Molnár
Rights and permissions
About this article
Cite this article
Słowik, R.K. Decay of the elements of the inverses of some triangular Toeplitz matrices. ActaSci.Math. 87, 541–550 (2021). https://doi.org/10.14232/actasm-021-028-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.14232/actasm-021-028-7