Polynomial Norm
For a polynomial
| [画像: P=sum_(k=0)^na_kz^k, ] |
(1)
|
several classes of norms are commonly defined. The l_p-norm is defined as
for p>=1, giving the special cases
||P||_1 = [画像:sum_(j)|a_k|]
(3)
||P||_2 = [画像: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
![画像: ||P||_(L_p)=[int_0^(2pi)|P(e^(itheta))|^p(dtheta)/(2pi)]^(1/p)](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/PolynomialNorm/NumberedEquation3.svg){kind=link}
for p>=1, giving the special cases
||P||_(L^1) = [画像:int_0^(2pi)|P(e^(itheta))|(dtheta)/(2pi)]
(7)
||P||_(L^2) = [画像: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 NormExplore with Wolfram|Alpha
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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 NormCite this as:
Weisstein, Eric W. "Polynomial Norm." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PolynomialNorm.html