Two-weight estimates for Fourier operators and Bernstein inequality

The norm estimation problem for Fourier operators acting from Lwp (\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{T}$$ \end{document}) to Lυq (\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{T}$$ \end{document}) where 1 < p ≦ q < ∞ was investigated. These results has been generalized to the two-dimensional case and applied to obtain generalizations of the Bernstein inequality for trigonometric polynomials of one and two variables. Also, the rates of convergence of Cesaro and Abel-Poisson means of functions f ∈ Lwp (\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{T}$$ \end{document}) has been estimated in the case p = q and υ ≡ w . The generalized Bernstein inequality applied to estimate the order of best trigonometric approximation of the derivative of functions f ∈ Lwp (\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{T}$$ \end{document}) in the space Lυq (\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{T}$$ \end{document}).

Best Trigonometric Approximation, Fractional Order Derivatives and Lipschitz Classes

Let C2π denote the space of 2π-periodic continuous functions and πn the set of trigonometric polynomials of degree ≦ n, where n ϵ P = {0, 1, … } . Given θ > 0, the well-known theorem of Stečkin and its converse state that the best approximation of an ƒ ϵ C2π with respect to the max-norm satisfies