Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 2 / b / Solution

The Fejér kernel is nonnegative, and preservation of constants givesBy evenness,Put $\delta =n^{-1/2}$. Use ${\omega }_2(f,|t|)\le (|t|/\delta +1{)}^2{\omega }_2(f,\delta )$ together with $F_n(t)\le n/2$ and $F_n(t)\le C/(n{t}^2)$. Splitting at $1/n$ and $\delta$ shows that the remaining weighted integral is uniformly bounded, so