By Topic

On the linear independence of a function and its derivatives

Sign In

Cookies must be enabled to login.After enabling cookies , please use refresh or reload or ctrl+f5 on the browser for the login options.

Formats Non-Member Member
$33 $13
Learn how you can qualify for the best price for this item!
Become an IEEE Member or Subscribe to
IEEE Xplore for exclusive pricing!
close button

puzzle piece

IEEE membership options for an individual and IEEE Xplore subscriptions for an organization offer the most affordable access to essential journal articles, conference papers, standards, eBooks, and eLearning courses.

Learn more about:

IEEE membership

IEEE Xplore subscriptions

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

We obtain the following results. 1) Suppose that z(t) and its first m derivatives z^{(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 to z^{(k)} and vanishes for 0 \leq k \leq m - 1 but does not vanish for k = m , then the vectors {z^{(k)}(t)} are linearly independent for each t in the domain of z(\cdotp) . 2) If now z^{(k)}(t), k = 0,...,m are mean-square continuous random processes such that z^{(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.

Published in:

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