An L 2 form of Bernstein's inequality

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Abstract
Suppose P is a pth degree real polynomial function in n variables and f=PǀSn-1 is the restriction of P to the unit sphere Sn-1 in Rn. Bernstein's inequality asserts that ([unk]0kf)2 + p2([unk]0k-1f)2 ≤ p2k ∥f∥∞2, where k ≥ 1 and differentiation is with respect to arc length θ along any geodesic in Sn-1. We find the constant corresponding to p2k when ∥f∥∞ is replaced by ∥f∥2. One application is a condition on the coefficients of the expansion in surface spherical harmonics of any g: Sn-1 → R, which condition suffices to assure that g is k times differentiable.

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