Applications of MVBV condition in \(L^1\) integrability
The authors introduce the following notion for a sequence of non negative real numbers, in order to guarantee \(L^1\) integrability of the associated sine (resp. cosine) series. A sequence \((a_n)\) is said to be a mean value bounded variation sequence (\(A=(a_n)\in MVBVS\)) if there exists \(\lambda\geq2\) and \(C(A)>0\) such that \[ \sum_{k=n}^{2n}| a_k-a_{k+1}| \leq {C(A)\over n}\sum_{k=[\lambda^{-1} n]}^{[\lambda n]}a_k \] for every \(n\geq1\). The following result generalizes former results (when \(A\) is decreasing to \(0\)): Assume that \(A\in MVBVS\). Then For \(0<\alpha<2\), \(x^{-\alpha}g(x)\sim x^{-\alpha} \sum_{n=1}^{\infty}a_n\sin(nx)\in L_{2\pi}\) if and only if \(\sum_{n=1}^{\infty}n^{1-\alpha}a_n<\infty \), for \(0<\alpha<1\), \(x^{-\alpha}f(x)\sim x^{-\alpha}\sum_{n=1}^{\infty}a_n\cos(nx)\in L_{2\pi}\) if and only if \(\sum_{n=1}^{\infty}n^{1-\alpha}a_n<\infty \), where \(L_{2\pi}\) stands for the space of \(2\pi\) periodic integrable functions. The case \(\alpha=0\) is partially treated in Theorem 3: Assume that \(A\in MVBVS\). Then \(g(x)\in L_{2\pi}\) implies that \(\sum_{n=1}^{\infty}n^{1-\alpha}a_n<\infty \). There exists \(A\in MVBVS\) such that \(\sum_{n=1}^{\infty}n^{1-\alpha}a_n<\infty \) but \(g(x)\notin L_{2\pi}\). They conjecture that under the assumption \(\sum_{k=n}^{2n}| a_k-a_{k+1}| \leq C(A)a_n\), we have \[ \sum_{n=1}^{\infty}n^{1-\alpha}a_n<\infty \Rightarrow g(x)\in L_{2\pi}. \]
scientific article; zbMATH DE number 5857783
scientific article