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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.


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".


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.