I've been given a problem about Walsh-Fourier series,tried to set formulas,used other theorems,but couldn't find a way out. Can anyone tell me hint?
The problem is as it states:
Prove that $$S_i(S_{2^s}f(x)) = \left\{ \begin{array}{c} S_{2^s}(f(x)), i>2^s\\ S_i(f(x)),i\le2^s \\ \end{array} \right. $$ where $S_{n}f(x)$ is Walsh-Fourier series $S_nf(x)=\sum_{k=0}^{n-1}\int_{[0,1]}fw_k$ and $w_k$ is Walsh function.If $k=\sum_{i=0}^{\infty}k_i2^i$ and $x=\sum_{j=0}^{\infty}x_j2^{-j}$ where $x_j \in \{0,1\}$ than $w_k(x)=(-1)^{\sum_{j=0}^{\infty}k_jx_j}$