Recipe

Compute a norm of a function on an interval

Use when: you're given a function and an interval and asked for \|f\|_1, \|f\|_2, or \|f\|_\infty

Goal: a single nonnegative number measuring the function's size, by the specific rule the requested norm uses

Taught in Norms and "energy" of a function

1. 1

   Decide whether the question wants \|f\|_1 (total absolute size), \|f\|_2 (energy-style size), or \|f\|_\infty (peak size): the three measure genuinely different things.

   Example

   For the square wave f(x)=1 on (0,\pi), f(x)=-1 on (-\pi,0), over [-\pi,\pi]: suppose all three norms are wanted, so every case below gets used.

   • \|f\|_1 or \|f\|_2 is requested — continue to Step 2 and build an integrand.
   • \|f\|_\infty is requested — skip the integral entirely and go straight to Step 4's supremum.

   Why: the three norms are defined by different formulas, so picking the wrong one changes every step that follows.

2. 2

   For \|f\|_1, the integrand is |f(x)|; for \|f\|_2, it's f(x)^2.

   Example

   For the square wave, |f(x)|=1 everywhere on [-\pi,\pi] since f only ever equals \pm1, and f(x)^2=1 everywhere too, for the same reason.

   ⚡ Shortcut

   When f(x)^2 (or |f(x)|) is even on a symmetric interval, halve the work: \int_{-b}^{b}f(x)^2\,dx=2\int_0^b f(x)^2\,dx. For f(x)=x^2 on [-1,1], f(x)^2=x^4 is even, so \int_{-1}^1 x^4\,dx=2\int_0^1x^4\,dx.

   Why: \|f\|_1 and \|f\|_2 are both defined by integrating one of these two specific functions of x, not f itself.

3. 3

   Compute the definite integral of the Step 2 integrand.

   Example

   \displaystyle\int_{-\pi}^{\pi}1\,dx=2\pi: the same integral serves both \|f\|_1 and \|f\|_2 here, since both integrands happened to equal 1.

   Why: this integral is the shared core of both the L^1 and L^2 definitions; only the last step tells them apart.

4. 4

   For \|f\|_1, the Step 3 integral is already the final answer. For \|f\|_2, take its square root. For \|f\|_\infty, skip the integral altogether and find \sup|f(x)| directly, usually by inspection of f's graph.

   Example

   \|f\|_1=2\pi\approx6.28; \|f\|_2=\sqrt{2\pi}\approx2.51; and since f never exceeds |\pm1|, \|f\|_\infty=1, read straight off the function's peak height.

   • \|f\|_1 — report the Step 3 integral unchanged.
   • \|f\|_2 — take the square root of the Step 3 integral.
   • \|f\|_\infty — find the largest value |f(x)| ever reaches on the interval, with no integration at all.

   Why: each norm's own definition specifies a different final operation on the integral, or no integral at all for \|f\|_\infty. Skipping that last operation reports the wrong quantity, such as energy \|f\|_2^2 instead of the norm \|f\|_2.