The nonlocal nature of the summability of Fourier series by certain absolute Riesz methods

Abstract

It is proved that for a large class of sequences { λ n } \{ {{\lambda _n}} \} the summability at a point of a Fourier series Σ A n ( t ) \Sigma A_n ( t ) by the absolute Riesz method | R , λ n , 1 | | {R,{\lambda _n},1} | is not a local property of the generating function. It is also proved, inter alia, that, for every ε > 0 \varepsilon > 0 , the | R , λ n , 1 | | {R,{\lambda _n},1} | summability of the factored series Σ A n ( t ) λ n − ε \Sigma A_n ( t )\lambda _n^{ - \varepsilon } at any point is always a local property of the generating function.