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Any relationship between coefficients and min/max of polynomial?

-1

$begingroup$

Is there any relationship between the coefficients $ c_0, c_1, dots, c_n $ of a polynomial $ f(x) $:

$$ f(x) = c_0 + c_1 x + dots + c_n x^n $$

and the min/max values of $ f(x) $ in, say, [0, 1] ?

Clearly, you can say that:

$$ f(x)_min leq f(0) = c_0 leq f(x)_max $$

$$ f(x)_min leq f(1) = c_0 + c_1 + dots + c_n leq f(x)_max $$

but what else restricts $ c_0, c_1, dots, c_n $ in terms of $ f(x)_min $ and $ f(x)_max $, except like... checking all the extrema of $ f(x) $ one by one?

In other words, if you want a polynomial $ f(x) $ of degree $ n $ that should fulfill $ L leq f(x) leq U $ where $ L $ and $ U $ are constants, what are the all possible values of $ c_0, c_1, dots, c_n $, i.e., domain of vector $ pmbc $ ?

polynomials

share|cite|improve this question

edited Mar 24 at 17:24

akai

asked Mar 24 at 16:44

akaiakai

1142

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It's not clear to me what your requirements are. As you note, you can find the true max/min using the coefficients if you're willing to do the computations. You can find many different types of bounds depending on the trade-off you make between simplicity and precision.
  $endgroup$
  – Michael Biro
  Mar 24 at 16:56
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MichaelBiro Thanks, edited! What I actually want to do is, as I've added as the last line, determine a polynomial f(x) that should be L < f(x) < U in some interval.
  $endgroup$
  – akai
  Mar 24 at 16:59
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That's still way too under-specified. Why can't I just take $f(x) = fracL + U2$? or $f(x) = L + (U - L)x$?
  $endgroup$
  – Michael Biro
  Mar 24 at 17:06
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MichaelBiro You're right. Edited again. I want to know what the all possible values of c0, c1, ..., cn i.e, domain of vector c.
  $endgroup$
  – akai
  Mar 24 at 17:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You can get tons of relations by using $L leq f(x)$ is equivalent to $f(x)-L geq 0$ and integrating that relation against any say continuous or even more generally integrable non-negative function on $[0,1]$ starting with $1$ but could use $x^a, a > -1$ etc; similarly the other way, using now $U-f(x) geq 0$. I would say it is hopeless to derive general conditions, but if you need something concrete in a given problem, the above ideas may help
  $endgroup$
  – Conrad
  Mar 24 at 17:42
  
  

  
  
  
  

 | 
show 3 more comments

-1

$begingroup$

Is there any relationship between the coefficients $ c_0, c_1, dots, c_n $ of a polynomial $ f(x) $:

$$ f(x) = c_0 + c_1 x + dots + c_n x^n $$

and the min/max values of $ f(x) $ in, say, [0, 1] ?

Clearly, you can say that:

$$ f(x)_min leq f(0) = c_0 leq f(x)_max $$

$$ f(x)_min leq f(1) = c_0 + c_1 + dots + c_n leq f(x)_max $$

but what else restricts $ c_0, c_1, dots, c_n $ in terms of $ f(x)_min $ and $ f(x)_max $, except like... checking all the extrema of $ f(x) $ one by one?

In other words, if you want a polynomial $ f(x) $ of degree $ n $ that should fulfill $ L leq f(x) leq U $ where $ L $ and $ U $ are constants, what are the all possible values of $ c_0, c_1, dots, c_n $, i.e., domain of vector $ pmbc $ ?

polynomials

share|cite|improve this question

edited Mar 24 at 17:24

akai

asked Mar 24 at 16:44

akaiakai

1142

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  It's not clear to me what your requirements are. As you note, you can find the true max/min using the coefficients if you're willing to do the computations. You can find many different types of bounds depending on the trade-off you make between simplicity and precision.
  $endgroup$
  – Michael Biro
  Mar 24 at 16:56
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MichaelBiro Thanks, edited! What I actually want to do is, as I've added as the last line, determine a polynomial f(x) that should be L < f(x) < U in some interval.
  $endgroup$
  – akai
  Mar 24 at 16:59
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  That's still way too under-specified. Why can't I just take $f(x) = fracL + U2$? or $f(x) = L + (U - L)x$?
  $endgroup$
  – Michael Biro
  Mar 24 at 17:06
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MichaelBiro You're right. Edited again. I want to know what the all possible values of c0, c1, ..., cn i.e, domain of vector c.
  $endgroup$
  – akai
  Mar 24 at 17:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You can get tons of relations by using $L leq f(x)$ is equivalent to $f(x)-L geq 0$ and integrating that relation against any say continuous or even more generally integrable non-negative function on $[0,1]$ starting with $1$ but could use $x^a, a > -1$ etc; similarly the other way, using now $U-f(x) geq 0$. I would say it is hopeless to derive general conditions, but if you need something concrete in a given problem, the above ideas may help
  $endgroup$
  – Conrad
  Mar 24 at 17:42
  
  

  
  
  
  

 | 
show 3 more comments

$begingroup$

Is there any relationship between the coefficients $ c_0, c_1, dots, c_n $ of a polynomial $ f(x) $:

$$ f(x) = c_0 + c_1 x + dots + c_n x^n $$

and the min/max values of $ f(x) $ in, say, [0, 1] ?

Clearly, you can say that:

$$ f(x)_min leq f(0) = c_0 leq f(x)_max $$

$$ f(x)_min leq f(1) = c_0 + c_1 + dots + c_n leq f(x)_max $$

but what else restricts $ c_0, c_1, dots, c_n $ in terms of $ f(x)_min $ and $ f(x)_max $, except like... checking all the extrema of $ f(x) $ one by one?

In other words, if you want a polynomial $ f(x) $ of degree $ n $ that should fulfill $ L leq f(x) leq U $ where $ L $ and $ U $ are constants, what are the all possible values of $ c_0, c_1, dots, c_n $, i.e., domain of vector $ pmbc $ ?

polynomials

share|cite|improve this question

edited Mar 24 at 17:24

akai

asked Mar 24 at 16:44

akaiakai

1142

$endgroup$

share|cite|improve this question

edited Mar 24 at 17:24

akai

asked Mar 24 at 16:44

akaiakai

1142

share|cite|improve this question

edited Mar 24 at 17:24

akai

asked Mar 24 at 16:44

akaiakai

1142

asked Mar 24 at 16:44

akaiakai

1142

asked Mar 24 at 16:44

akaiakai

1142