Methods of trigonometric approximation and generalized smoothness. I

We give a unified approach to trigonometric approximation and study its interrelation with smoothness properties of functions. In the first part our concern lies on convergence of the Fourier means, interpolation means and families of linear trigonometric polynomial operators in the scale of the $L_p$-spaces with $0<p\le+\infty$. We establish a general convergence theorem which allows to determine the ranges of convergence for approximation methods generated by classical kernels.
The second part will deal with the equivalence of the approximation errors for families of linear polynomial operators generated by classical kernels in terms of $K$-functionals generated by homogeneous functions and general moduli of smoothness. It will also be shown that the results of the classical approximation theory on the Fourier means and interpolation means in the case $1\le p\le+\infty$, classical differential operators and moduli of smoothness are direct consequences of our general approach.