### Quick description

Suppose that you have a function that has been written as an infinite sum , where is some sequence of "nice" functions. Often it is possible to find by finding a linear map from the set of functions you are interested in to or such that . Then . In the language of linear algebra, it is possible in many natural problems of this kind to identify a dual basis to the basis .

### Prerequisites

Basic real and complex analysis.

### Example 1

Let . Suppose you know that for almost every , with . Then

### Example 2

Let be a holomorphic function defined on some domain that includes the origin, and suppose that can be expanded in a power series on . Suppose that is some closed curve in 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 round a closed curve that winds once round the origin is ) tell us that

### 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

Fri, 08/05/2009 - 02:41 — 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 - 08:11 — gowersI 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 aspect of the idea to the article itself.