On Boundedness of Lagrange Interpolation
نویسنده
چکیده
We estimate the distribution function of a Lagrange interpolation polynomial and deduce mean boundedness in Lp; p < 1: 1 The Result There is a vast literature on mean convergence of Lagrange interpolation, see [4{ 8] for recent references. In this note, we use distribution functions to investigate mean convergence. We believe the simplicity of the approach merits attention. Recall that if g : R! R, and m denotes Lebesgue measure, then the distribution function mg of g is mg( ) := m (fx : jg(x)j > g) ; 0: (1) One of the uses of mg is in the identity [1,p.43] k g kpLp(R)= Z 1 0 pt mg(t)dt; 0 < p <1: (2) Moreover, the weak L1 norm of g may be de ned by k g kweak(L1)= sup >0 mg( ): (3)
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