Generalized absolute Hausdorff summability of orthogonal series

Abstract

For 1≦k≦2 and a sequence γ:={γ(n)}n=1∞ that is quasi β-power monotone decreasing with β > 1-1/k, we prove the {pipe}A,γ{pipe}k summability of an orthogonal series, where A is either a regular or Hausdorff matrix. For β > -3/4, we give a necessary and sufficient condition for {pipe}A,γ{pipe}k summability, where A is Hausdorff matrix. Our sufficient condition for β > -3/4, is weaker than that of Kantawala [1], β > -1/k, for {pipe}E,q,γ{pipe}k summability; and of Leindler [4], β>-1 for {pipe}C,α,γ{pipe}k, α < 1/4. Also, our result generalizes the result of Spevakov [6] for {pipe}E,q,1{pipe}1 summability. © 2013 Akadémiai Kiadó, Budapest, Hungary.