ON DIVERGENCE OF FOURIER-WALSH SERIER

Abstract

Let the sequences \big \{beta_k\big \} ^\infty_{n=1}$ be fixed such that $\lim\limits_{k\rightarrow\infty}(M_{2k}-M_{2k-1})=+\infty, \beta_k>0, $\lim\limits_{k\rightarrow\infty} \beta_k=0.$ In this paper, we prove that there exists a function $f_0(x) \in L^1_{[0,1]}$ such that the Fourier series of the function $f_0(x)$ with respect to the subsystem $ \big \{W_{n_k}(x) $ \big \}_{k=1}^\infty=\big \{W_m(x):M_{2s-1}\leq m\leq M_{2s}, s= 1,2,... \big \}$ diverges in the $L^1_{[0,1]}$ metric, and the Fourier-Walsh coefficients satisfy the condition $\sum\limits_{k=1}^\infty |a_{n_k}|\beta_{n_k}<\infty.$