On universal Fourier series in the Walsh system
The article is dedicated to the study of the existence of universal functions whose Fourier series with respect to the Walsh system are universal in the class of all measurable functions in the sense of signs. For this end, let \(c_k(U)=\int_a^bU(x)\varphi_k(x)dx\) be the Fourier coefficients of the function \(U\in L^1[a,b]\), with respect to an orthonormal system \(\Phi=\{\varphi_k\}_k\) defined on the interval \([a,b]\) and let \(S_m(U,\varphi)(x)=\sum_{k=1}^mc_k(U)\varphi_k(x)\) be the partial sums. Therefore, a function \(U\in L^1[a,b]\) for the space \(S\) with respect to the system \(\Phi=\{\varphi_k\}_k\) is universal in the sense of signs if for each \(f\in S\), we can find a sequence of signs \(\{\delta_k\}_k\) with \(\delta_k=\pm 1\) for \(k\geq 1\), for which the series \(\sum_{k\geq 1} \delta_kc_k(U)\varphi(x)\) converges to \(f(x)\) in \(S\). Then the principal result is the following theorem. Theorem. There exist a sequence \(\{ N_m\}_{m\geq 1}\) of positive integers and \(U\in L^1[0,1]\) whose Fourier-Walsh series has monotone decreasing coefficients and converges in the norm of \(L^1[0,1]\) such that: \par (a) Given \(f\in M[0,1)\), we can find \(\delta_k=\pm 1\) for \(k\geq 1\), such that \[ \lim_{N_m\to \infty}\sum_{k=0}^{N_m}\delta_kc_k(U)W_k(x)=f(x) \] almost everywhere on \([0,1)\) \par (b) \(U\) is universal for \(M[0,1)\) (in the case of convergence in measure) in the Walsh system in the sense of signs. Remark. \par (i) There is a typo on page 868, line 30, first equation; \[ \lim_{k\to \infty}\int_a^b|f_k(x)-f(x)|dx=0\text{ instead of } \lim_{k\to \infty}\int_0^1|f_k(x)-f(x)|dx. \] \par (ii) There is a typo on page 869, line 1, \[ A=\{x\in[a,b), |f(x)|<\infty\}\text{ and }A^{\pm}=\{x\in[a,b), f(x)=\pm\infty\} \] instead of \[ A=\{x\in[0,1), |f(x)|<\infty\}\text{ and }A^{\pm}=\{x\in[0,1), f(x)=\pm\infty\} \]
scientific article; zbMATH DE number 7596648
scientific article