Bound for quotient of eigenvalues
- From: José Carlos Santos <jcsantos@xxxxxxxx>
- Date: Fri, 24 Apr 2009 15:05:45 +0100
Hi all,
For each real-analytic function _f_ from [0,1] into itself, consider the
2x2 real matrix
a b
b c
where a = int(x^2 f(x),dx), b = int(x f(x),dx) and c= int(f(x),dx) (all
integrals are from 0 to 1). Unless _f_ is the null function, this matrix
is positive definite and therefore it has two positive eigenvalues.
Consider the quotient e_1/e_2 where e_1 is the greatest eigenvalue and
e_2 is the smallest one. My question is: is there an upper bound for
the quotients obtained by this method? My guess is that the answer is
negative, but I was unable to prove it.
Best regards,
Jose Carlos Santos
.
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