Add and subtract something nearby that is simpler

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Add and subtract something nearby that is simpler

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[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
[math] f(x) = g(x) + (f(x)-g(x)).[/math]

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.<comment thread="233" /> 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. $h := f-g$.  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

[note article incomplete] More examples wanted. [/note]

[EXAMPLE]

A classic "$\varepsilon/3$" example: show that if a sequence of continuous functions $f_n: [a,b] \to \R$ converges uniformly to a limit $f: [a,b] \to \R$, 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 $|f(x)-f(x_0)| \leq \varepsilon$.

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]

Let $f \in L^p(\R)$ for some $1 \leq p < \infty$, and let $\phi_n: \R \to \R^+$ 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 $f*\phi_n(x) := \int_\R f(x-y) \phi_n(y)\ dy$ converge in $L^p$ to $f$.

(Supply proof here)

[EXAMPLE]

(Calderon-Zygmund theory)

[EXAMPLE]

(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]].

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Add and subtract something nearby that is simpler

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