Some new estimates useful in applications concerning the approximation of certain classes of functions characterized by the generalized continuity modulus from the space \(L_{2}(\mathcal {S}^{m-1})\) , \(m\ge 3\) , by partial spherical sums of Fourier-Laplace series are obtained. More precisely, the proofs of two theorems on this issue, which can be directly applied to solving particular problems of mathematical physics, approximation theory, etc., are presented. For this purpose, we use a generalized spherical shift defined by Rudin in (Trans. Amer. Math. Soc. 68, 287–303 1950) as a translation operator on the unit sphere.
