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2011/2012 Moments and the Range of the Derivative
Eugen J. Ionascu, Richard Stephens
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Real Anal. Exchange 37(1): 129-146 (2011/2012).


In this note we introduce three problems related to the topic of finite Hausdorff moments. Generally speaking, given the first \(n+1\) (\(n\in \mathbb N\cup \{0\}\)) moments, \(\alpha_0\), \(\alpha_1\),..., \(\alpha_n\), of a real-valued continuously differentiable function \(f\) defined on \([0,1]\), what can be said about the size of the image of \(\frac{df}{dx}\)? We make the questions more precise and we give answers in the cases of three or fewer moments and in some cases for four moments. In the general situation of \(n+1\) moments, we show that the range of the derivative should contain the convex hull of a set of \(n\) numbers calculated in terms of the Bernstein polynomials, \(x^k(1-x)^{n+1-k}\), \(k=1,2,...,n\), which turn out to involve expressions just in terms of the given moments \(\alpha_i\), \(i=0,1,2,...n\). In the end we make some conjectures about what may be true in terms of the sharpness of the interval range mentioned before.


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Eugen J. Ionascu. Richard Stephens. "Moments and the Range of the Derivative." Real Anal. Exchange 37 (1) 129 - 146, 2011/2012.


Published: 2011/2012
First available in Project Euclid: 30 April 2012

zbMATH: 1250.44004
MathSciNet: MR3016855

Primary: 26A04% , 44A60%
Secondary: 26A05

Keywords: derivative , moments , quadratic and linear functions , Spline

Rights: Copyright © 2011 Michigan State University Press

Vol.37 • No. 1 • 2011/2012
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