Behavior of the sequence of norms of primitives of a function
The behavior of the sequence of the \(L_p\)-norm of primitives of a function is characterized by its spectrum, being the smallest closed set outside of which the Fourier transform \(\hat{f}\) vanishes. More precisely, let \(1 \leq p \leq \infty\) and \((I^nf)_{n=0}^{\infty} \subset L_p(\mathbb{R})\) be the sequence of primitives of \(f\), then it is proved that \( \lim_{n \rightarrow \infty}\|I^n f\|_p^{1/n} = \sigma^{-1}\), where \(\sigma = \inf\{ |\xi| : \xi \in \text{supp}\hat{f} \}\).
scientific article; zbMATH DE number 5726582