Approximation by Fourier Means and Generalized Moduli of Smoothness

The quality of approximation by Fourier means generated by an arbitrary generator with compact support in the spaces $L_p$, $1\le p\le\nobreak +\infty$, of $2\pi$-periodic $p$th integrable functions and in the space $C$ of continuous $2\pi$-periodic functions in terms of the generalized modulus of smoothness constructed from a $2\pi$-periodic generator is studied. Natural sufficient conditions on the generator of the approximation method and values of smoothness ensuring the equivalence of the corresponding approximation error and modulus are obtained. As applications, Fourier means generated by classical kernels as well as the classical moduli of smoothness are considered.