Re: What is the meaning of this pattern of polynomial coefficients?

[Robert A. Macy]

Group,

Have more to add.  After judiciously scaling the ordinate,
to some interim value, making the ordinate go between 0.5
and up to around 2; then the high and low coefficient
magnitudes become EXACT mirror images of themselves.  

For example, pretend the minus sign is really a 180 phase
shift, then the polynomial always takes the form of

  for n being odd...
A*x^n - B*x^(n-1) + C*x$(n-2) ... - C*x^2 + B*x - A

  and for n being even...
A*x^n - B*x^(n-1) + C*x$(n-2) ... + C*x^2 - B*x + A

for 2 < n < 17

plus one more feature...
The center values dominate with the center 
coefficient located at round(n/2)+1 being the maximum value

If I plot the log of the magnitudes of the coefficients, it
looks exactly like an inverted parabola with the peak in
the center index.  

The original data is derived from a physical observation
and I did not expect to see such symmetry.

Is there a better curve fit than a polynomial for this
family of curves? 

       - Robert -

On Mon, 31 Jan 2005 11:43:31 -0800
 "Robert A. Macy" <address@hidden> wrote:
> Group,
> 
> I’ve got a matrix of complex values.  
> 
> Each row relates to one variable.
> 
> Along each row relates to an ordinate (vector).
> 
> After doing a polynomial fit for each row whether 2, 3,
> 4,
> 5, … are used for the number of coefficients; the
> characteristics of those coefficients look almost
> identical!  Except for every other one is shifted in
> phase
> by 180 degrees and their magnitude is different, but
> their
> “shape” in the complex plane is almost identical. 
> 
> What is the significance of this?  
> 
> - Robert –
> 
> 
> 
> 
>
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