Generalized absolute Hausdorff summability of orthogonal series

For 1≦k≦2 and a sequence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma :={\{\gamma(n)\}}_{n=1}^{\infty}$\end{document} that is quasi β-power monotone decreasing with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\beta>1-\frac{1}{k}}$\end{document}, we prove the |A,γ|k summability of an orthogonal series, where A is either a regular or Hausdorff matrix. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\beta>-\frac{3}{4}}$\end{document}, we give a necessary and sufficient condition for |A,γ|k summability, where A is Hausdorff matrix. Our sufficient condition for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\beta>-\frac{3}{4}}$\end{document} is weaker than that of Kantawala [1], \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\beta>-\frac{1}{k}}$\end{document} for |E,q,γ|k summability; and of Leindler [4], β>−1 for |C,α,γ|k, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\alpha<\frac{1}{4}}$\end{document}. Also, our result generalizes the result of Spevakov [6] for |E,q,1|1 summability.