Re: Bound for quotient of eigenvalues

On 24-04-2009 16:11, achille wrote:

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.

Consider f(x) = (x(1-x))^(m-1),

a = B(m+2,m) = (m+1)/(2*(2m+1)) B(m,m)
b = B(m+1,m) = 1/2 B(m,m)
c = B(m,m)

where B(x,y) is the beta function. Notice

B(m,m)^(-1) [ a b ] = [ 1/4 +O(1/m) 1/2 ]
[ b c ] [ 1/2 1 ]

converges to a singular matrix as m -> \infinity.
It is easy to see for this f(x), e_1/e_2 ~ O(m)
for large m.

Thanks a lot for your help.

Best regards,

Jose Carlos Santos
.

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