Szasz's theorem and its generalizations (Q2071778)

The author investigates generalizations of a theorem of Szasz, given by the estimate \[ \int_{{\mathbb{R}^n}} |\xi|^{\theta p} |\mathcal{F} f(\xi)|^p d\xi \le c \, \|f\|_{\dot{A}^s_{r,q}{\mathbb{R}^n}}\, , \] where \(\mathcal{F}\) is the Fourier transform, \(c\) a general constant depending only on the parameters \(s,p,q,r,n, \theta\) and \(\dot{A}^s_{r,q}{\mathbb{R}^n}\) denotes either the homogeneous Besov space \(\dot{B}^s_{r,q}(\mathbb{R}^n)\) or the homogeneous Triebel-Lizorkin space \(\dot{F}^s_{r,q}(\mathbb{R}^n)\). His main result consists in an if and only if assertion for the validity of the above inequality.