# A bound for Mean values of Fourier transforms

Authors
Type
Published Article
Publication Date
May 16, 2011
Submission Date
May 16, 2011
Source
arXiv
We show that there exists a sequence $\{n_k, k\ge 1\}$ growing at least geometrically such that for any finite non-negative measure $\nu$ such that $\hat \nu\ge 0$, any $T>0$, $$\int_{-2^{n_k} T}^{2^{n_k} T} \hat \nu(x) \dd x \ll_\e T\,2^{2^{(1+\e)n_k}} \int_\R \Big|{\sin {xT} \over xT} \Big|^{n_k^2} \nu(\dd x).$$