Wavelet and Linear Algebra

Abstract  In this paper, the space  $ c (I) $  is introduced and some of its properties examined. Then with the help of a diameter norm  on the space  $ c_{0}(I)$, a norm is defined on the space $ c (I) $ called as D- norm, which  is an extension of  the $ d-$norm. It is also shown that the D- norm is equivalent to the supremum norm.  The extreme points of the unit ball of the spaces $ c_{0}(I) $  and  $ c (I) $ are also specified. In addition we find  some orthogonal vectors in the space $ c (I) $.

Abstract Chi-Kwang Li, Bit-Shun Tam and Nam-Kiu Tsing have obtained necessary and sufficient condition  for a linear operator on linear space of generalized doubly stochastic matrices to be strong preserver of doubly stochastic matrices and permutation matrices.
    We show if a linear operator $T:M_m\rightarrow M_n$ is a (strong) preserver of doubly stochastic matrices, then $T$ is a (strong) preserver of the linear manifold of r-generalized doubly stochastic matrices and the linear space of generalized doubly stochastic matrices. Also we give necessary and sufficient condition for a linear operator $T:M_m\rightarrow M_n$ to be (strong) preserver of doubly stochastic matrices and permutation matrices.