An L form of Bernstein's inequality.
Suppose P is a pth degree real polynomial function in n variables and f=PS(n-1) is the restriction of P to the unit sphere S(n-1) in R(n). Bernstein's inequality asserts that ([unk](0) (k)f)(2) + p(2)([unk](0) (k-1)f)(2) </= p(2k) parallelf parallelinfinity(2), where k >/= 1 and differentiation is with respect to arc length theta along any geodesic in S(n-1). We find the constant corresponding to p(2k) when parallelf parallelinfinity is replaced by parallelf parallel(2). One application is a condition on the coefficients of the expansion in surface spherical harmonics of any g: S(n-1) --> R, which condition suffices to assure that g is k times differentiable.
Backus
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GE
Institute of Geophysics and Planetary Physics, University of California at San Diego, A-025, La Jolla, California 92093.