217 Absolute-Value-to-the-p
A function obtained by raising the absolute value of another function to a fixed parameter power that is used to build norms and integrals that measure size.
definition [d] (\(|f|^{p}\)) The nonnegative function obtained by raising the absolute value of \(f\) to the power \(p\).
where
- \(f\) is a function.
- \(|f|\) is the absolute value of \(f\).
- \(p\) is a real exponent with \(p \geq 1\).
- \(|f|^{p}\) is the pointwise \(p\)-th power of \(|f|\).
- the scalar field of values of \(f\) may be \(\mathbb{R}\).
- the scalar field of values of \(f\) may be \(\mathbb{C}\).
Note:
- the integral of \(|f|^{p}\) being finite is the membership condition for \(L^{p}\).
217.1 Elementary Example
217.1.1 Simple
The map \(f \mapsto |f|^{p}\) raises the absolute value to the power \(p\). On three sample points with \(p = 1\), this is just \(|f|\).
\[ A = \{ 1,\ 2,\ 3 \},\quad p = 1 \]
\[ f(1) = -2,\quad f(2) = 0,\quad f(3) = 4 \]
\[ |f|^{1}(1) = 2,\quad |f|^{1}(2) = 0,\quad |f|^{1}(3) = 4 \]
where
- \(f\) is the function.
- \(|f|^{p}\) is the pointwise \(p\)-th power of \(|f|\).
217.1.2 General
For \(p = 2\), square the absolute values on a longer sample.
\[ A = \{ 0,\ 1,\ 2,\ 3 \},\quad p = 2 \]
\[ f(0) = -1,\quad f(1) = 2,\quad f(2) = -3,\quad f(3) = 0 \]
\[ |f|^{2}(0) = 1,\quad |f|^{2}(1) = 4,\quad |f|^{2}(2) = 9,\quad |f|^{2}(3) = 0 \]
where
- finiteness of \(\int |f|^{p}\) is the \(L^{p}\) membership condition.