Expansion coefficients of size 10^100,000

Since writing An Applied Mathematician’s Apology I find myself noting more examples of how unquantitative mathematicians may be.

Here’s one. A well-known theorem by Müntz asserts that, for example, the function f(x) = x can be approximated arbitrarily closely for 0< x<1 by linear combinations of the functions 1, x2, x4, x6, … What Müntz’s theorem doesn’t tell you is that to do this, say to 6-digit accuracy, you’ll need 140,000 of those terms with coefficients as large as 10100,000 ! Such approximations would be useless in any conceivable application.

You might think that the discovery that a theorem is useless for any application would be of interest to mathematicians. In fact, most would regard this effect as just a curiosity. It doesn’t touch the essential truth or beauty of the theorem, and indeed, perhaps it enhances it by shining a light on the power of rigorous proof.

(I asked a leading expert in the area whether he knew of this 10100,000 effect. No, he responded, “I am not a numbers man.”)

In other words, a mainstream view among mathematicians is that it is not their business to care whether or not a theorem is capable of being applied. This raises the question, if it is not their business, then whose business is it? Engineers? I imagine most mathematicians would feel that no, one need not go so far as that. It is the business of the numerical analysts, who are, after all, mathematicians of a kind.

[27 July 2022]

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