Absolute Nörlund summability factors for Fourier series (Q5902831)

Let \(f\in L(-\pi,\pi)\) be \(2\pi\)-periodic with \(f(0)=0\) and with Fourier series \(\sum^{\infty}_{n=1}A_ n(x)\) at the point x. For \(t>0\), define \[ \phi (t):=1/2[f(x+t)+f(x-t)],\phi_ 1(t)=t^{- 1}\int^{t}_{0}\phi (u)du,R(t)=\phi (t)-\phi_ 1(t). \] The first theorem (1) proved is the following: suppose that \(0\leq p_ n\downarrow\), \(P_ n=p_ 0+...+p_ n\), \(P_ n\sum^{\infty}_{k=n}[kP_ k\log (k+2)]^{-1}=O[(\log \log n)^ c]\) for some \(c\geq 1\); then in order that \(\sum^{\infty}_{n=1}A_ n(x)/[\log (n+1)]\) should be absolutely Nörlund summable \(| N,p_ n|\) whenever \(R(t)(\log \log 1/t)^ c\in BV(0,\pi),\) it is necessary and sufficient that \(c>1\). Moreover, the factor \(\log (n+1)\) in the series cannot be replaced by \([\log (n+1)]^{\beta}\) \((0<\beta <1)\). This relates to, generalizes, and formulates necessary restrictions on the order of the summability factor in, a number of earlier results. A second Theorem replaces \(| N,p_ n|\) by the absolute Cesàro- Nörlund iteration \(| (C,1)(N,p_ n)|\).