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On the linear independence of a function and its derivatives

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1 Author(s)
Brandenburg, L. ; Bell Labs, Murray Hill, NJ, USA

We obtain the following results. 1) Suppose thatz(t)and its firstmderivativesz^{(k)}(t), k = 1,...,m, are continuous functions with values in a normed linear vector space. We define a class of linear functionals and show that if a functional in the class is applied toz^{(k)}and vanishes for0 leq k leq m - 1but does not vanish fork = m, then the vectors{z^{(k)}(t)}are linearly independent for eachtin the domain ofz(cdotp). 2) If nowz^{(k)}(t), k = 0,...,mare mean-square continuous random processes such thatz^{(m+1)}(cdotp)has a nonvanishing white-noise component, then the random variables{z^{(k)}(t)}, k = 0,..,m, are linearly independent. These results are shown to be related both in formulation and method of solution.

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Automatic Control, IEEE Transactions on  (Volume:18 ,  Issue: 6 )