Invertibility of a tridiagonal operator with an application to a non-uniform sampling problem

Abstract

Let T be a tridiagonal operator on which has strict row and column dominant property except for some finite number of rows and columns. This matrix is shown to be invertible under certain conditions. This result is also extended to double infinite tridiagonal matrices. Further, a general theorem is proved for solving an operator equation using its finite-dimensional truncations, where T is a double infinite tridiagonal operator. Finally, it is also shown that these results can be applied in order to obtain a stable set of sampling for a shift-invariant space.

Keywords:

• Diagonal dominance
• finite-dimensional truncation
• LU-decomposition
• shift-invariant space
• stable set of sampling
• tridiagonal operator

AMS Subject Classifications:

• Primary: 47B37
• Secondary: 15A60
• 94A20

Acknowledgements

We thank the referee for meticulously reading the manuscript and giving us several valuable suggestions in improving the initial version of the manuscript. We also thank Prof. S. H. Kulkarni and Y .V. S. S. Sanyasiraju for some fruitful discussions and their useful suggestions.

Notes

No potential conflict of interest was reported by the authors.