I'v read somewhere that one motivation for Hardy to define his maximal function is the game of cricket. But I can't see how they are related. Could anyone provide some more information on their connections?
[Math] Cricket and the Hardy-Littlewod maximal function
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Best Answer
One hesitates to explain a joke, but this is quite a nice joke, and I cannot resist answering a cricket question (especially now). Suppose a batsman scores 20, 100, 30, 40, 70 and 0 in his last six innings (0 being the most recent). Being upset at scoring 0 in his last innings, he might say to himself -- at least I am averaging 35 in my last two innings; or going further, that I'm averaging 36.67 in my last three; or averaging 35 in my last four; or 48 in my last five; or 43.33. Probably he would most prefer the fifth statement which gives the largest average, or the most satisfaction.
Suppose now the batsman does this over an entire season of cricket, after each innings computing his satisfaction until then. Call the total satisfaction the sum of the satisfactions after each innings in the season. Then for a given stock of scores in a season, the Hardy-Littlewood maximal theorem gives that the batsman's total satisfaction is a maximum if his scores are in descending order throughout the series -- or in other words, his satisfaction is a maximum precisely when his batting has been in decline throughout the season!
There is a nice discussion of this in Bela Bollobas's problem book "The art of mathematics: coffee time in Memphis" -- see problem 85 on Satisfied Cricketers: the Hardy-Littlewood maximal theorem there.
The English cricket team has clearly taken this Theorem a little too seriously!