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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.

Note iconIncomplete 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".

Comments

This title seems kind of

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

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.