A Direct Theorem of Approximation Theory for a General Modulus of Smoothness

We introduce the notion of general modulus of smoothness in the spaces $L_p$ of $2\pi$-periodic $p$th-power integrable functions; in these spaces, the coefficients multiplying the values of a given function at the nodes of the uniform lattice are the Fourier coefficients of some $2\pi$-periodic function called the generator of the modulus. It is shown that all known moduli of smoothness are special cases of this general construction. For the introduced modulus, in the case $1 \le p \le {+\infty}$, we prove a direct theorem of approximation theory (a Jackson-type estimate). It is shown that the known Jackson-type estimates for the classical moduli, the modulus of positive fractional order, and the modulus of smoothness related to the Riesz derivative are its direct consequences. We also obtain a universal structural description of classes of functions whose best approximations have a certain order of convergence to zero.