question about polynomials

[fel...@fsmat.at]

hi,

i have a problem. I want to know, if for a given (even) natural number
n there exists
a polynomial a_1+a_2z+...+a_nz^{n-1} whose coefficients fulfill

(1) a_1+...+a_n = 1/4
(2) |a_1|+|a_3|+...+|a_{n-1}| < 1/4
(3) |a_2|+|a_4|+...+|a_n| < 1/4
(4) 2 ( |a_1+a_2|+|a_3+a_4|+...+|a_{n-1}+a_n| ) >= 1/sqrt(2).

I have come along this problem in the course of a work i am doing on
manifold valued wavelet analysis and would be very happy, if such
coefficients didn't exist!

greetings, philipp

[alex....@gmail.com]

=============
Verify if the polynomial
P(z)=(z^{n-1}+..+z+1)/(4n) = (z^n -1)/(4n(z-1)) ,
that is when a_1=a_2=...=a_n=1/(4n) ,
satisfies conditions (1)--(4).
Here n is even , n>2 .

[Patrick Coilland]

fel...@fsmat.at nous a récemment amicalement signifié :

I gave you an answer yesterday on sci.maths and you ignored it and ask
now on this group !

Take :

Let e be any small positive rational number such that
1-8e >= 1/sqrt(2)
1/4 >> n e

Then :
a_1 = 1/4 - n/2 e
a_2 = 1/8 - (n-4)/2 e
a_3 = e
a_4 = -1/8 + e
Others = e

works.

--
Patrick

[alainv...@yahoo.fr]

fel...@fsmat.at a écrit :

Patrick Coilland gave a clear answer in
sci.math forum this day ,

Alain

[rena...@aol.com]

The following is a counterexample:
n=4
a1=a2=-1/16+epsilon, a3=a4=3/16-epsilon,
where epsilon is small.

[Peter Spellucci]

In article <ea83iq$qmu$1...@news.ks.uiuc.edu>,

for n=2 this is clearly impossible but for n=4 I found

a1=a2=(1/(4*sqrt(2)) + 1/8)/2 a3=a4=-(1/(4*sqrt(2))-1/8)/2

satisfying all your constraints
sorry
peter