A Quick Intro to the Maximal Function

based on one by its creator, G.H.Hardy


Suppose that Mighty Casey makes 10 hits in his first game of the baseball season. He plays once a week for nine more weeks, but scores 0 in games 2 thru10. I'll call his scores "f(1)", "f(2)" etc and I'll label the whole set of them as a function "f". This is the red bar graph you'll see in the applet below.

Suppose Casey's friends want to know how he's doing after each game. After game 1 he says his average is 10. After game 2 he says his average is 5, etc. These numbers are called Mf(1), Mf(2) etc. They form a maximal function "Mf", which is the green bar graph below.

After this intro, you can change one of Casey's scores to see what happens. For example, if he scores 6 runs in game 5, he will report the 6 that week (so, Mf(5)=6) and the next week he will report his average for weeks 5 and 6 only to make himself look as good as possible (so, Mf(6)=3).

You will notice that Mf is "bigger" than f. How can we measure that? A "norm" is a way to measure the size of a function, and there are at least three standard norms we can use:

The 1-norm is the total area. If you haven't played with the applet yet, the red area comes from just the first rectangle, which is 1x10, so the 1-norm of f is 10. The 1-norm of Mf is the green area, which is much bigger. You can click on "Calculate norms" to see how much bigger. Note: my applet rounds all fractions off to integers.

The 2-norm is the square root of the sum of the heights squared (a calculation similar to finding the length of a vector). Originally it is 10 for f, and about 12 for Mf. This norm is probably the most interesting one in mathematics.

The infinity-norm is the tallest height. For f, this is originally 10. The infinity norm of Mf will always equal that of f. It's called the "infinity"-norm because it fits a pattern made by the 2-norm, 3-norm , etc (I won't go into the details).

One central question is, "How much bigger can Mf be than f?" Think this over as you play with the applet below (which should have loaded by the time you read this far!)

Notice that you can change f by left-clicking with your mouse (anywhere in the lower left box). This automatically changes Mf. You can choose which norm you want to use, and then click "Calculate" to see the results.

You have probably noticed that Mf cannot be bigger than f if we use the infinity norm, but it can be a lot bigger if we use the 1-norm. [It can be a zillion times bigger if we extend the baseball season long enough (because the harmonic series diverges)]. Hardy proved that Mf can be only "a little bigger" than f using the 2-norm, no matter how long the season is!

There are interesting ways to define variations on Mf in higher dimensions. These can lead to very hard open problems. Mf and its variants are useful in analysis and advanced differential equations. I hope you enjoyed the math and the applet!