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Polynomial Norm


For a polynomial

 P=sum_(k=0)^na_kz^k,
(1)

several classes of norms are commonly defined. The l_p-norm is defined as

 ||P||_p=(sum_(k=0)^n|a_k|^p)^(1/p)
(2)

for p>=1, giving the special cases

||P||_1=sum_(j)|a_k|
(3)
||P||_2=sqrt(sum_(k)|a_k|^2)
(4)
||P||_infty=max_(k)|a_k|.
(5)

Here, |P|_infty is called the polynomial height. Note that some authors (especially in the area of Diophantine analysis) use |P| as a shorthand for ||P||_infty and ||P|| as a shorthand for ||P||_2, while others (especially in the area of computational complexity) used |P| to denote the l^2-norm ||P||_2 and (Zippel 1993, p. 174).

Another class of norms is the L^p-norms, defined by

 ||P||_(L_p)=[int_0^(2pi)|P(e^(itheta))|^p(dtheta)/(2pi)]^(1/p)
(6)

for p>=1, giving the special cases

||P||_(L^1)=int_0^(2pi)|P(e^(itheta))|(dtheta)/(2pi)
(7)
||P||_(L^2)=sqrt(int_0^(2pi)|P(e^(itheta))|^2(dtheta)/(2pi))
(8)
||P||_(L^infty)=sup_(|z|=1)|P(z)|
(9)

(Borwein and Erdélyi 1995, p. 6).


See also

Bombieri Norm, Matrix Norm, Norm, Unit Circle, Vector Norm

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References

Borwein, P. and Erdélyi, T. "Norms on P_n." §1.1.E.3 in Polynomials and Polynomial Inequalities. New York: Springer-Verlag, pp. 6-7, 1995.Zippel, R. Effective Polynomial Computation. Boston, MA: Kluwer, 1993.

Referenced on Wolfram|Alpha

Polynomial Norm

Cite this as:

Weisstein, Eric W. "Polynomial Norm." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PolynomialNorm.html

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