On Banach spaces of absolutely and strongly convergent Fourier series (Q1813370)

For \(\lambda\geq 1\), the complex sequence \(\{s_ k\}^ \infty_{k=0}\) is said to be strongly convergent of index \(\lambda\) (strong \(\lambda\)) to \(t\) provided \[ (n+1)^{-1} \sum^ \infty_{k=0} | (k+ 1)(s_ k- t)- (s_{k-1}- t)|^ \lambda= o(1)\quad\text{as } n\to\infty, \] and absolutely convergent of index \(\lambda\) (absolute \(\lambda\)) to \(t\) if \(\{s_ k\}^ \infty_{k=0}\) has limit \(t\) and \(\sum^ \infty_{k=0} k^{\lambda-1}| s_ k- s_{k-1}|^ \lambda< \infty\). Restricting themselves to continuous, \(2\pi\)-periodic, complex functions \(f\) with Fourier partial sums \(\{s_ n f\}^ \infty_{n=0}\), the authors study the classes \({\mathcal L}^ \lambda= \{f: s_ nf\to f\) strong \(\lambda\) uniformly\}. They show that \({\mathcal L}^ \lambda\) is not a Banach space under \(\| f\|_ u= \sup_ n\| s_ n f\|\) but is under four other (all equivalent) norms each expressed in terms of the Fourier coefficients of \(f\) and \(\| f\|_ u\). One of them is \[ \sup_ n\Bigl\|\Bigl({1\over n+1} \sum^ n_{k=0} k^ \lambda | s_ k f- s_{k-1} f|^ \lambda\Bigr)^{1/\lambda}\Bigr\| +\| f\|_ u. \] Further, they show that \({\mathcal A}^ \lambda\subset {\mathcal L}^ \lambda\) properly and \({\mathcal L}^ \lambda\) is not a Banach space under any of the four norms for \({\mathcal L}^ \lambda\) but is under each of four other norms (also all equivalent), one of them being \(| \widehat f_ c(0)|/2+ \sum^ \infty_{k=1} |\hat f_ c(k)|+ |\widehat f_ s(k)|\), where \(\widehat f_ s\) and \(\widehat f_ c\) are the coefficients of the Fourier and Cesàro-1 series for \(f\).