Add and subtract something nearby that is simpler

Quick description

Suppose one is trying to estimate an expression involving a complicated function $f(x)$ (e.g. something like $\int_Y |\int_X f(x) K(x,y)\ dx|^p\ dy$ ). But, one knows or believes that $f(x)$ is somehow "close" (at least "on average") to a simpler quantity $g(x)$ . Then it can often become advantageous to add and subtract $g(x)$ from $f(x)$ , i.e. to substitute

$f(x) = g(x) + (f(x)-g(x)).$

In many cases the term involving $g(x)$ is the "main term", and one can take advantage of the simpler structure of $g$ to continue estimating this portion of the expression. Meanwhile, the term $f(x)-g(x)$ is often an "error term"; either $f-g$ is already small, in which case one can hope that the total contribution of this term to the expression one wants to estimate is already small, or $f-g$ exhibits some sort of cancellation which will also make the final contribution to the original expression small (e.g.^◊ by the trick "use integration by parts to exploit cancellation").

For more complicated expressions (e.g. bilinear, multilinear, or nonlinear expressions) it is often useful to give the error term $f-g$ its own name, e.g. $= f-g$ . Then $f = g+h$ , and any multilinear expression involving one or more copies of $f$ will split into a "main term" involving all $g$ 's, plus lots of "error terms" involving one or more $h$ 's. Often, one treats the error terms by relatively crude upper bound estimates, but works carefully to estimate the main term as accurately as possible.

Typical examples of choices of $g(x)$ include

the value $f(x_0)$ of $f$ at a point $x_0$ nearby to $x$
the average value $\bar{f} = \frac{1}{|B|} \int_B f$ of $f$ on some suitable set $B$ ;
the conditional expectation ${\Bbb E}(f|{\mathcal B})(x)$ of $f$ with respect to some $\sigma$ -algebra ${\mathcal B}$ .
Some sort of regularization, discretization, or other approximation to $f$ (e.g. one could convolve $f$ with an approximation to the identity).

Prerequisites

Undergraduate real analysis

This article is incomplete. More examples wanted.

Example 1

A classic " $\varepsilon/3$ " example: show that if a sequence of continuous functions $[a,b] \to \R$ converges uniformly to a limit $[a,b] \to \R$ , then $f$ is also continuous.

To prove this, pick an $x_0 \in [a,b]$ and $\varepsilon > 0$ . The task is to show that if $x$ is sufficiently close to $x_0$ , then $|f(x)-f(x_0)| \leq \varepsilon$ .

But by hypothesis, we expect $f(x)$ to be close to $f_n(x)$ , and $f(x_0)$ to be close to $f_n(x_0)$ , for $n$ large. Adding and subtracting these terms, and using the triangle inequality, we are led to the bound

$|f(x)-f(x_0)| \leq |f(x)-f_n(x)| + |f_n(x)-f_n(x_0)| + |f_n(x_0)-f(x_0)|.$

Because of the uniform convergence, we know that for $n$ large enough (independent of $x$ or $x_0$ - this is important!), we can ensure that $|f(x)-f_n(x)| \leq \varepsilon/3$ and $|f_n(x_0)-f(x_0)| \leq \varepsilon/3$ . Once we pick such an $n$ , we can then use the fact that $f_n$ is continuous to conclude that $|f_n(x)-f_n(x_0)| \leq \varepsilon/3$ for $x$ close enough to $x_0$ , and the claim follows.

Example 2

Let $f \in L^p(\R)$ for some $1 \leq p < \infty$ , and let $\R \to \R^+$ be a sequence of approximations to the identity (thus $\int_\R \phi_n(x)\ dx = 1$ , and $\lim_{n \to \infty} \int_{|x|>\varepsilon} \phi_n(x) = 0$ for all $\varepsilon > 0$ . Show that the convolutions $= \int_\R f(x-y) \phi_n(y)\ dy$ converge in $L^p$ to $f$ .

(Supply proof here)

Example 3

(Calderon-Zygmund theory)

Example 4

(Roth's argument for three-term APs)

General discussion

In some cases, particularly those involving integration by parts or substitution, one wishes to use multiplying and dividing $f(x) = \frac{1}{g(x)} g(x) f(x)$ instead of adding and subtracting. For instance, given an integral involving an expression $e^{-x^2}$ , one may wish to multiply and divide by $2x$ in order to set up either an integration by parts, or a substitution $y = x^2$ .

A more advanced version of this technique is generic chaining.

Comments

Inline comments

The following comments were made inline in the article. You can click on 'view commented text' to see precisely where they were made.

Strange elaboration

Thu, 23/04/2009 - 01:08 — JoseBrox

"either $f-g$ is already small, in which case one can hope that the total contribution of this term is already small" sounds a bit strange to me! :D

When I saw the title of this,

Thu, 23/04/2009 - 09:40 — gowers

When I saw the title of this, I thought it might be an amusing article that gave a surprisingly advanced perspective on addition and subtraction, but now I see that it is doing something else. I wonder if a more specific title such as "Add and subtract something simpler" might be an improvement: I think that captures more what the article is about, and also sticks in the mind as a slogan, which is something I hope will happen a lot with the Tricki. (In general, I prefer titles in the form of commands, though I haven't always managed to come up with such titles myself.)

Later: I did in the end go ahead and change the title.

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