Polynomial Bar Norm

For $p=\sum a_j z^j$, define

$$\begin{eqnarray*} \vert\vert P\vert\vert _1 &=& \int_0^{2\pi} \vert P(e^{i\theta... ...vert\vert _\infty &=& {\rm max}_{\vert z\vert=1} \vert P(z)\vert \end{eqnarray*}$$

$$\begin{eqnarray*} \vert P\vert _1 &=&\sum_j \vert a_j\vert\\ \vert P\vert _2 &=... ..._j\vert^2}\\ \vert P\vert _\infty&=&{\rm max}_j \vert a_j\vert, \end{eqnarray*}$$

where the $\vert\vert P\vert\vert _i$ norms are functions on the Unit Circle and the $\vert P\vert _i$ norms refer to the Coefficients $a_0$, ..., $a_n$.

See also Bombieri Norm, Norm, Unit Circle

References

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, p. 151, 1989.