From polynomials to Chebyshev polynomials












2














I was wondering how they got from the polynomial to a Chebyshev polynomial as outlined here:




In order to obtain “strong” stability, we replace the condition ($2.7$) by
$$left|prod_{j=1}^N (1-tau_jlambda)right| leq K qquad forall lambda in [mu,lambda_max], tag{2.8}$$ where $mu$ is some number in the interval $(0, lambda_min]$, and $K$ is some number $0<K<1$. The problem of finding the “optimal” values for the $tau_j$’s can then be reformulated as




Find $tau_1$, $tau_2$, $ldots$, $tau_N$ such that $p_N(lambda) =prod_{j=1}^N (1-tau_jlambda)$ satisfies
begin{array}{l@{qquad}l}
|p_N(lambda)| leq K quad forall lambda in [mu, lambda_max] && text{(STABILITY),} \
displaystyle |p'_N(0)| =sum_{j=1}^N tau_j quad text{maximal} && text{(OPTIMALITY).}
end{array}




Using the remarkable optimality properties of the Chebyshev polynomials $T_N(cdot)$ of degree $N$, Markoff [$5$] ($1892!$), we have that if $K$ is given by $$K=1/T_N left(frac{lambda_max +mu}{lambda_max -mu}right)$$




Anyone knows?



The actual paper, I'm referring to is here.










share|cite|improve this question





























    2














    I was wondering how they got from the polynomial to a Chebyshev polynomial as outlined here:




    In order to obtain “strong” stability, we replace the condition ($2.7$) by
    $$left|prod_{j=1}^N (1-tau_jlambda)right| leq K qquad forall lambda in [mu,lambda_max], tag{2.8}$$ where $mu$ is some number in the interval $(0, lambda_min]$, and $K$ is some number $0<K<1$. The problem of finding the “optimal” values for the $tau_j$’s can then be reformulated as




    Find $tau_1$, $tau_2$, $ldots$, $tau_N$ such that $p_N(lambda) =prod_{j=1}^N (1-tau_jlambda)$ satisfies
    begin{array}{l@{qquad}l}
    |p_N(lambda)| leq K quad forall lambda in [mu, lambda_max] && text{(STABILITY),} \
    displaystyle |p'_N(0)| =sum_{j=1}^N tau_j quad text{maximal} && text{(OPTIMALITY).}
    end{array}




    Using the remarkable optimality properties of the Chebyshev polynomials $T_N(cdot)$ of degree $N$, Markoff [$5$] ($1892!$), we have that if $K$ is given by $$K=1/T_N left(frac{lambda_max +mu}{lambda_max -mu}right)$$




    Anyone knows?



    The actual paper, I'm referring to is here.










    share|cite|improve this question



























      2












      2








      2







      I was wondering how they got from the polynomial to a Chebyshev polynomial as outlined here:




      In order to obtain “strong” stability, we replace the condition ($2.7$) by
      $$left|prod_{j=1}^N (1-tau_jlambda)right| leq K qquad forall lambda in [mu,lambda_max], tag{2.8}$$ where $mu$ is some number in the interval $(0, lambda_min]$, and $K$ is some number $0<K<1$. The problem of finding the “optimal” values for the $tau_j$’s can then be reformulated as




      Find $tau_1$, $tau_2$, $ldots$, $tau_N$ such that $p_N(lambda) =prod_{j=1}^N (1-tau_jlambda)$ satisfies
      begin{array}{l@{qquad}l}
      |p_N(lambda)| leq K quad forall lambda in [mu, lambda_max] && text{(STABILITY),} \
      displaystyle |p'_N(0)| =sum_{j=1}^N tau_j quad text{maximal} && text{(OPTIMALITY).}
      end{array}




      Using the remarkable optimality properties of the Chebyshev polynomials $T_N(cdot)$ of degree $N$, Markoff [$5$] ($1892!$), we have that if $K$ is given by $$K=1/T_N left(frac{lambda_max +mu}{lambda_max -mu}right)$$




      Anyone knows?



      The actual paper, I'm referring to is here.










      share|cite|improve this question















      I was wondering how they got from the polynomial to a Chebyshev polynomial as outlined here:




      In order to obtain “strong” stability, we replace the condition ($2.7$) by
      $$left|prod_{j=1}^N (1-tau_jlambda)right| leq K qquad forall lambda in [mu,lambda_max], tag{2.8}$$ where $mu$ is some number in the interval $(0, lambda_min]$, and $K$ is some number $0<K<1$. The problem of finding the “optimal” values for the $tau_j$’s can then be reformulated as




      Find $tau_1$, $tau_2$, $ldots$, $tau_N$ such that $p_N(lambda) =prod_{j=1}^N (1-tau_jlambda)$ satisfies
      begin{array}{l@{qquad}l}
      |p_N(lambda)| leq K quad forall lambda in [mu, lambda_max] && text{(STABILITY),} \
      displaystyle |p'_N(0)| =sum_{j=1}^N tau_j quad text{maximal} && text{(OPTIMALITY).}
      end{array}




      Using the remarkable optimality properties of the Chebyshev polynomials $T_N(cdot)$ of degree $N$, Markoff [$5$] ($1892!$), we have that if $K$ is given by $$K=1/T_N left(frac{lambda_max +mu}{lambda_max -mu}right)$$




      Anyone knows?



      The actual paper, I'm referring to is here.







      polynomials numerical-methods chebyshev-polynomials






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




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      edited Dec 26 at 3:04









      Rócherz

      2,7612721




      2,7612721










      asked Feb 19 '15 at 12:45









      unrealbot

      111




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