On Quantities of the Type of Modulus of Continuity and Analogs of $K$-Functionals in the Spaces $S^{(p,q)}(\sigma^{m-1})$

The paper continues the research of the author begun in 2003–2021. Quantities of the type of modulus of continuity of functions defined on the sphere in the space $S^{(p,q)}(\sigma^{m-1})$ are studied. These quantities are generated by a family of operators of multiplier type. Their equivalence to analogs of $K$-functionals is established.