Majorants and uniqueness of series in the Franklin system
For any \(g\in L^1[0,1]\) and \(y>0\), let \(\lambda_g(y):= \{x\in[0,1]:|g(x)|>y\}\) represent the distribution function of \(g\). For any Franklin series \(S\), let \(S^*:=\sup_{N>0}|S_N|\) represent the maximal function associated with its sequence of partial sums. The author has shown [Mat. Zametki 46, No. 2, 51-58 (1989; Zbl 0707.42020)] that \(S\) is the Franklin-Fourier series of some \(f\in L^1[0,1]\) if and only if \(S_N\to f\) almost everywhere as \(N\to\infty\) and \(\liminf_{y\to\infty} y\lambda_{S^*}(y)=0\). Here, the author shows that if a Franklin series \(S\) converges almost everywhere to some function \(f\) and \(y\lambda_{S^*}(y)\to 0\) as \(y\to\infty\), then \(f\) is A-integrable and \(S\) is the ``A-integral'' Franklin-Fourier series of \(f\). The author also shows that if \(\sum a_nf_n\) is the Franklin-Fourier series of some \(f\in L^1[0,1]\) and \(\{\varepsilon_n\}\) is any bounded sequence of numbers, the \(\sum\varepsilon_na_nf_n\) is the ``A-integral'' Franklin-Fourier series of some A-integrable function \(g\). The multidimensional versions of these results are also discussed.
scientific article; zbMATH DE number 1028539
scientific article