Envelope (waves)

In mathematics, a radially unbounded function is a function $f:{\mathbb{ℝ}}^n\to \mathbb{ℝ}$ for which

$\Vert x\Vert \to \mathrm{\infty }\Rightarrow f(x)\to \mathrm{\infty }.$

Such functions are applied in control theory. Notice that the norm used in the definition can be any norm defined on ${\mathbb{ℝ}}^n$, and that the behavior of the function along the axes does not necessarily reveal that it is radially unbounded or not; i.e. to be radially unbounded the condition must be verified along any path that results in:

$\Vert x\Vert \to \mathrm{\infty }$

For example the functions

$f_1(x)=(x_1-x_2{)}^2$

$f_2(x)=(x_1^2+x_2^2)/(1+x_1^2+x_2^2)+(x_1-x_2{)}^2$

are not radially unbounded since along the line $x_1=x_2$, the condition is not verified even though the second function is globally positive definite.

References

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