Unconditional convergence of the differences of Fej´er kernels on L2 (R)

Abstract

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 convolution of $f$ 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 unconditionally for all $f\in L^2(\mathbb{R})$. Let $(n_k)$ be a lacunary sequence, and $\{c_k\}_{k=1}^\infty \in \ell^\infty$. Define
$$\mathcal{R}f(x)=\sum_{k=1}^\infty c_k\left(\sigma_{n_{k+1}}f(x)-\sigma_{n_k}f(x)\right).$$ 
Then  there exists a constant $C>0$ such that
$$\|\mathcal{R}f\|_2\leq C\|f\|_2$$
for all $f\in L^2(\mathbb{R})$, i.e., $\mathcal{R}f$ is of strong type $(2,2)$. As a special case it follows that $\mathcal{G}f$ also is of strong type $(2,2)$.