Let \(S^n\) be the \(n\)-dimensional sphere in \(\mathbb R^n\) and \(A=\{U_j,\psi_j\}_{j=1}^m\) be an atlas for \(S^n\), i.e. open sets \(U_j\subset S^n\) cover \(S^n\), \(\psi_j\) are homeomorphic from \(U_j\) to the unit ball \(B(0,1)\subset \mathbb R^n\) and \(\psi_i\circ \psi_j^{-1}\) are \(C^{\infty}\) on \(\psi_j(U_j\cap U_i)\). Let \(\{\chi_j\}_{j=1}^m\) be the partition of unity with respect to \(\{U_j\}_{j=1}^m\). For \(f:S^n\to \mathbb R\) we introduce \(\pi_j(f)(x)=f\circ \psi_j^{-1}(x)X_{B(0,1)}(x)\), where \(X_E\) is the indicator of \(E\). Then \[ W^{\tau}_p(S^n)=\Biggr\{f\in L_p(S^n): \| f| W^{\tau}_p(S^n)\| = \biggl(\sum^m_{j=1}\| \pi_j(\chi_jf)| W^{\tau}_p(\mathbb R^n)\| ^p\biggr)^{1/p}m+n/p\) if \(p>1\) or \(k\geq m+n/p\) if \(p=1\). Theorem 3.3. Suppose that the assumption above holds, \(X\subset S^n\) is finite and \(h_{X,S^n}\) is sufficiently small. If \(u\in W^{\tau}_p(S^n)\) satisfies \(u| X=0\), then \[ \| u| W^{m}_q(S^n)\| \leq Ch^{\tau-m-n(1/p-1/q)_+}\| u| W^{\tau}_p(S^n)\| . \] Applications to estimates of approximation of \(u\) by solutions of some least-square problems are given.