Is the approximation of a function by its Fejér means monotone? (Q915044)

Let \(T=[-,)\) and \(f\in L_ p(T)\), \(1\leq p\leq \infty\), where in the case \(p=\infty\), it is assumed that f is continuous on T. Let \(K_ n\) be the nth Fejér kernel and \(K_ n*f\) the convolution. Here it is shown that while the approximation sequence \(\{\| f-K_ n*f\|_ p\}\) is not monotone in general, there does exist some sort of monotonicity property. It is proved that for every \(n>m\), \[ \| f-K_ n*f\|_ p\leq (2+(m/n))^{| (2/p)-1|}\| f-K_ m*f\|_ p. \] That \(\{\| f-K_ n*f\|_ p\}\) is not monotone, is demonstrated for the case \(p=\infty\), and, at the proof-stage of the paper, it is added that L. de Michele has obtained an extensions of this result for other values of p, \(1\leq p<\infty\), \(p\neq 2\).