Some Problems in Fourier Analysis and Matrix Theory

We discuss some problems studied in diverse contexts but with a common theme; the use of Fourier analysis to evaluate norms of som $\mathrm{e} $ special matrices. Let $\ovalbox{\tt\small REJECT}_{n} $ be the space of $n\mathrm{x} $ $n $ matrices. For $A\in\ovalbox{\tt\small REJECT}_{n} $ let $||A|| = \sup\{||Ax| | : x\in \mathbb{C}^{n})||x||=1\} $, be the usual operator norm of $A $. Let $A\circ X $ be the entrywise product of two matrices A and $X $ and let $||A||_{S} = \sup\{||A\circ X| | : ||X||=1\} $. This is the norm of the linear map on $\ovalbox{\tt\small REJECT}_{n} $ defined as $X\mapsto A\circ X $. Since $A\circ X $ is a principal submatrix of $A\otimes X $ , we have $||A\circ X||\leq||A\otimes X||=$ $||A||||X|| $ , and hence $||A||_{S}\leq||A|| $