To find the value of a coefficient, do something that kills all the other terms

Quick description

Suppose that you have a function $f$ that has been written as an infinite sum $f(x)=\sum_{n=1}^\infty a_nf_n(x)$ , where $f_1,f_2,\dots$ is some sequence of "nice" functions. Often it is possible to find $a_n$ by finding a linear map $\phi_n$ from the set $F$ of functions you are interested in to $\R$ or $\C$ such that $\phi_n(f_m)=\delta_{mn}$ . Then $\phi_n(f)=a_n$ . In the language of linear algebra, it is possible in many natural problems of this kind to identify a dual basis to the basis $f_1,f_2,\dots$ .

This article is incomplete. This is a bit short. It could do with more examples, including finite-dimensional ones, and some general discussion.

Prerequisites

Basic real and complex analysis.

Example 1

Let $[0,2\pi)\rightarrow\C$ . Suppose you know that $f(x)=\sum_{n=-\infty}^\infty a_ne^{inx}$ for almost every $x$ , with $\sum_{n=-\infty}^\infty|a_n|^2<\infty$ . Then

$\frac 1{2\pi}\int_0^{2\pi}f(x)e^{-imx}dx=\frac 1{2\pi}\int_0^{2\pi}\sum_{n=-\infty}^\infty a_ne^{inx}e^{-imx}dx=\frac 1{2\pi}\sum_{n=-\infty}^\infty \frac{a_n}{2\pi}\int_0^{2\pi}e^{i(n-m)x}dx=\sum_{n=-\infty}^\infty a_n\delta_{nm}=a_m.$

Example 2

Let $f$ be a holomorphic function defined on some domain $D$ that includes the origin, and suppose that $f$ can be expanded in a power series $f(z)=\sum_{n=0}^\infty a_nz^n$ on $D$ . Suppose that $C$ is some closed curve in $D$ that winds once around the origin. Basic results in complex analysis (a function with an antiderivative integrates to zero round any closed curve, and the integral of $z^{-1}$ round a closed curve that winds once round the origin is $2\pi i$ ) tell us that

$\frac 1{2\pi i}\int_Cz^{-(m+1)}f(z)dz=\frac 1{2\pi i}\int_C\sum_{n=0}^\infty a_nz^{n-m-1}dz=\sum_{n=0}^\infty \frac{a_n}{2\pi i}\int_Cz^{n-m-1}dz=\sum_{n=0}^\infty a_n\delta_{nm}=a_m.$

Example 3

The Lagrange interpolation formula can be derived in this way; details are at "Use basic examples to calibrate exponents".

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Comments

This title seems kind of

Fri, 08/05/2009 - 01:41 — Anonymous (not verified)

This title seems kind of long, even for Tricki.

How about "To find the value of a coefficient, do something that kills all other terms"

I suppose I was worried that

Fri, 08/05/2009 - 07:11 — gowers

I suppose I was worried that some pedant might say that making everything zero kills all other terms. But I've decided not to be worried by that after all and leave the $\delta_{mn}$ aspect of the idea to the article itself.