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Let $K_n(x)$ denote the Fej\'er kernel given by$$K_n(x)=\sum_{j=-n}^n\left(1-\frac{|j|}{n+1}\right)e^{-ijx}$$ and let$\sigma_nf(x)=(K_n\ast f)(x)$, where as usual $f\ast g$ denotes the convolutionof $f$ and $g$. Let the sequence $\{n_k\}$ be lacunary. Then the series$$\mathcal{G}f(x)=\sum_{k=1}^\infty\left(\sigma_{n_{k+1}}f(x)-\sigma_{n_k}f(x)\right)$$ converges unconditionallyfor all $f\in L^2(\mathbb{R})$. Let $(n_k)$ be a lacunary sequence, and$\{c_k\}_{k=1}^\infty \in \ell^\infty$. Define$$\mathcal{R}f(x)=\sum_{k=1}^\inftyc_k\left(\sigma_{n_{k+1}}f(x)-\sigma_{n_k}f(x)\right).$$ Then there exists aconstant $C>0$ such that $$\|\mathcal{R}f\|_2\leq C\|f\|_2$$ for all $f\inL^2(\mathbb{R})$, i.e., $\mathcal{R}f$ is of strong type $(2,2)$. As a specialcase it follows that $\mathcal{G}f$ also is of strong type $(2,2)$.

Unconditional convergence of the differences of Fej\'er kernels on $L^2(\mathbb{R})$