Absolute Nevanlinna summability and Fourier series (Q1583955)

Given a series \(\sum u_n\) let \(F(\omega)= \sum_{n< \omega} u_n\). Let \(q_\delta\equiv q_\delta(t)\) be defined for \(0\leq t<1\). The \(N(q_\delta)\) transform \(N(F,q_\delta)\) of \(F\) is defined by \[ N(F,q_\delta)(\omega)= \int^1_0 q_\delta(t) F(\omega t) dt. \] The series \(\sum u_n\) is said to be summable \(|N(q_\delta)|\) if \(N(F,q_\delta)(\omega)\in \text{BV}(A,\infty)\), for some \(A\geq 0\). In this interesting paper the author proves a theorem on \(|N(q_\delta)|\) summability of Fourier series and two theorems on absolute summability factors of Fourier series. A number of results on absolute Cesàro summability are deduced.