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
.
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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 - 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 — 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.