I'm trying to do an exercise in Pinsky's "Introduction to Fourier analysis and wavelets":
Suppose that $f$ satisfies $L^{2}$ Holder condition with $\alpha=1$. Prove that $\sum_{n\in \mathbb{Z}} |n|^{2}|\hat{f}(n)|^{2}<\infty$.
The author suggests applying Fatou's lemma to the fomula: $$||f_{h}-f||_{2}^{2}=\sum_{n\in \mathbb{Z}}|e^{inh}-1|^{2}|\hat{f}(n)|^{2},$$ here $||f_{h}-f||_{2}^{2}=\int_{0}^{2\pi}|f(x+h)-f(x)|^{2}dx$ and the $L^{2}$ Holder condition means $||f_{h}-f||_{2}^{2}\le Kh^{\alpha}$. However I don't know how to use this hint. Can anyone help me? Thanks a lot.
I think you have stated Holder continuity wrongly. In the definition you have $\|f_h-f\|_2 \leq Kh^{\alpha}$ . (No square on the left). Hence We have $\sum |\frac {e^{inh}-1} h|^{2} |\hat {f} (n)|^{2} \leq K$. Now apply Fatou's Lemma. [$\frac {e^{inh}-1} h \to in$ as $h \to 0$].