I. Fourier analysis and the dot-product

Metonymy is everywhere in mathematics, but a good metaphor is hard to find. This one and the next are two of my favorites.

In the simplest form of Fourier analysis, the calculation of the Fourier
sine series for an odd function *f*(*x*) defined on
,
the function
is represented
as a linear combination of the functions
etc.

where the coefficients *b*_{1}, *b*_{2}, etc. are given by the integrals

The metaphier. In , a vector can be represented as a linear combination of the basis elements :

where the coefficients *b*_{1}, *b*_{2}, etc. are given by the dot products

This works because the basis
is *orthonormal*:

The metaphor: in the vector space of piecewise-smooth, odd functions on , define

Then the functions
are
orthonormal with respect to this dot-product, as can be checked by
carrying out the integrals. So writing the Fourier sine series for *f*
is exactly analogous to writing **v** as a linear combination of
using the dot-products as coefficients.

(For more details see a text like David Powers, *Boundary Value
Problems* Academic Press, New York 1972 or Walter Rudin,
*Principles of Mathematical Analysis* McGraw Hill, New York 1974.)

- 1. Metonymy and metaphor
- 2. How to recognize
mathematical metonymy
- 3. How to recognize
a mathematical metaphor
- 4. Metaphors from modern mathematics:

I. Fourier analysis and the dot-product - 5. Metaphors from modern mathematics:

II. The Jordan normal form and the structure of abelian groups

© copyright 1999, American Mathematical Society.