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Suppose all geodesics of two Riemannian metrics g and $$\bar g$$ defined on a (connected, geodesically complete) manifold M n coincide. At each point x ∈ M n , consider the common eigenvalues ρ 1, ρ2, ... , ρn of the two metrics (we assume that ρ1 ≥ ρ2 ≥ ⋯ ρn) and the numbers $$\lambda _i = \left( {\rho _1 \rho _2 \cdot \cdot \cdot \rho _n } \right)^{{1 \mathord{\left/ {\vphantom {1 {\left( {n + 1}...
In the Sobolev space $$W_p^k (\Omega )$$ , where Ω is a bounded domain in ℝn with a Lipschitzian boundary, for an arbitrarily given $$m \in \mathbb{N}$$ , we construct a basis such that the error of approximation of a function $$W_p^k (\Omega )$$ the Nth partial sum of its expansion with respect to this basis can be estimated in terms of the modulus of smoothness $$\omega m(D^k...
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