DocumentCode
813325
Title
On the linear independence of a function and its derivatives
Author
Brandenburg, L.
Author_Institution
Bell Labs, Murray Hill, NJ, USA
Volume
18
Issue
6
fYear
1973
fDate
12/1/1973 12:00:00 AM
Firstpage
661
Lastpage
663
Abstract
We obtain the following results. 1) Suppose that 
and its first 
derivatives 
, 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 
and vanishes for 
but does not vanish for 
, then the vectors 
are linearly independent for each 
in the domain of 
. 2) If now 
are mean-square continuous random processes such that 
has a nonvanishing white-noise component, then the random variables 
, are linearly independent. These results are shown to be related both in formulation and method of solution.
and its first derivatives , 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 and vanishes for but does not vanish for , then the vectors are linearly independent for each in the domain of . 2) If now are mean-square continuous random processes such that has a nonvanishing white-noise component, then the random variables , are linearly independent. These results are shown to be related both in formulation and method of solution.Keywords
Functional analysis; Vector spaces; Aerodynamics; Automatic control; Feedback; Humans; Leg; Random processes; Random variables; Stability; Vectors; Vehicle dynamics;
fLanguage
English
Journal_Title
Automatic Control, IEEE Transactions on
Publisher
ieee
ISSN
0018-9286
Type
jour
DOI
10.1109/TAC.1973.1100431
Filename
1100431
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