Necessary and sufficient conditions for the imbedding of some classes of functions

Necessary and sufficient conditions are found for membership of a function $f(x)\in L^p(0,1)$ in the class $L^q\Phi (L)$, $1\le p\le q<\infty$. A connection is established between the modulus of continuity $\omega(\delta,f^*)$ and, on the one hand, the Fourier coefficients $a_n(f^*)$ and $b_n(f^*)$, and on the other hand the modulus of continuity $\omega(\delta,f)$.