On the norms of discrete analogues of convolution operators

Abstract:

We consider a discrete analogue of convolution operator $T(f) = K*f$ from $L^p({\mathbb {R}}^d)$ to $L^p({\mathbb {R}}^d)$: $T_{dis}(g) = K_{dis}* g$ from $\ell ^p(\mathbb {Z}^d)$ to $\ell ^p(\mathbb {Z}^d)$ where $K_{dis} = K \vert _{\mathbb {Z}^d}$ and $\hat K$ is supported in the fundamental cube. We show that the estimate $\|T_{dis}\|_p \le C^d \|T\|_p$ with $C > 1$ cannot be improved for a certain range of $p$.