Some approximation theorems (Q1395916)

Let \(L^2(S^n)\) denote the usual Lebesgue space over the unit sphere \(S^n\) with respect to the surface area measure of \(S^n\); \(x_0\) be a fixed point in \(\mathbb{R}^n\). Let \(P(x_0, m)\) denote the space of all harmonic polynomials vanishing at \(x_0\) together with all their derivatives of order less than or equal to \(m\), and \(m> 0\). It is known that \(P(x_0, m)\) is dense in \(L^2(S^n)\) if and only if \(|x_0|\geq 1\). The author addresses the following questions: a) Does the above result remains valid if \(L^2(S^n)\) is replaced by any \(L^p(S^n)\) with \(p> 2\)? b) Could \(S^n\) be replaced by more general surfaces?