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i suppose it's fairly well known that if a (continuous, real-valued) function $f$ on the real line satisfies

$f(x-y)=f(x)-f(y)+const$

then it is necessarily linear.

are there any general characterizations of "approximate" linearity? for example, what can be said if $|f(x-y)-f(x)+f(y)-f(0)|$ is bounded by some small $\epsilon$? or, more relevant to what i need, if the $L^2$ norm of this difference is bounded by a small constant? in particular, suppose,

$$E[(f(X-Y)-f(Y)+f(Y)-f(0))^2]\leq\epsilon$$

for a pair of independent random variables $X,Y$. is there a sense in which $f$ is approximately linear?

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1 Answer 1

up vote 11 down vote accepted

Let $E$ and $E'$ be Banach spaces. Mappings $f:E\to E'$, which satisfy the inequality $$\|f(x + y) − f(x) − f(y)\| \leq\epsilon$$ for all $x, y \in E$, are called $\epsilon$-additive (or approximately additive). The main result concerning approximately additive functions is due to D. Hyers (link).

Theorem. Let $f(x)$ be an $\epsilon$-additive mapping of a Banach space $E$ into another Banach space $E'$. Then $l(x)=\lim f(2^nx)/2^n$ exists for each $x$ in $E$, $l(x)$ is linear, and the inequality $$\|f(x)-l(x)\|\leq\epsilon$$ is true for all $x$ in $E$. Moreover, $l(x)$ is the only linear mapping satisfying this inequality. If $f(x)$ is continuous at at least one point, then $l(x)$ is continuous everywhere in $E$.

So in your case $E=L^2(\Omega)$, $E'=\mathbb R$.

Edit. As Yemon Choi indicated, finite dimensional versions of the result had been discovered earlier and independently. Check out, for instance, the Pólya and Szegö problem book (Ch 3, Problem 99):

Assume that the terms of the sequence $a_1,a_2,a_3,\dots$ satisfy the condition $$a_m+a_n-1 < a_{m+n} < a_m+a_n+1.$$ Then $$\lim\limits_{n\to\infty}\frac{a_n}{n}=\omega$$ exists; $\omega$ is finite and we have $$\omega n-1 < a_n < \omega n +1.$$

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The case $E=E'= {\mathbb R}$ seems to have been (re)discovered independently several times - I think I've seen it in some edition of Lang's Algebra, where it's attributed to an observation of John Tate, for instance. –  Yemon Choi Jul 6 '10 at 20:04
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Thanks for the comment. I don't know the history of this problem very well but you are probably right. There is even a 1D version of this result for sequences in "Problems and theorems in Analysis" by Pólya and Szegö ! –  Andrey Rekalo Jul 6 '10 at 20:21
    
thanks, both for the useful references. i found a bunch of interesting related results and generalizations, including numerous papers on what is apparently called the "Hyers–Ulam–Rassias stability problem." in my original answer i seem to have over-simplified what i needed and i still can't find an appropriate reference. the problem i really have is the following: assuming $\|f(x)+f(s-x)-g(s)\|<\epsilon$, is it then possible to say something about the function $f$? (ideally, that it's in some sense approximately additive.) thanks again! -yiannis –  Yiannis Jul 8 '10 at 20:12
    
Yo're welcome. You might want to start another MO question to get more attention of the MO community. –  Andrey Rekalo Jul 8 '10 at 20:40
    
Thanks, I'll do that -- –  Yiannis Jul 8 '10 at 20:49

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