Fourier Approximation ## Fourier Approximation

## Fourier Approximation

1997 marks the 175-th anniversary of the publication, in 1822, of Joseph Fourier's

*Théorie analytique de la chaleur (The Analytical Theory of Heat*), in which he advanced his idea of approximating functions on a closed interval, or, alternatively, periodic functions on the real line, by trigonometric polynomials. (He had been working on this idea since at least 1807.) For example, a function

*f(x)* defined on the interval [-1,1] has a Fourier series (as we now call it)

Since the time of Fourier, the question of convergence of Fourier series has been intensively studied.

Consider, for example, the function

*f(x)=x(x*^{2}-1)(4x^{2}-1)(9x^{2}-4)(16x^{2}-9) on the interval [-1,1].

This is an "odd" function, i.e., *f(-x) = -f(x)*. For such a function, each *a*_{k} = 0, so its Fourier series has only sine terms. The following images show the graphs of *f(x)* and some of its Fourier approximants. Here the n-th Fourier approximant *s*_{n}(x) to *f(x)* is the sum of the first n terms of its Fourier series,

Graph of *f(x)* and its first, second, and third Fourier approximants.

Graph of

*f(x)* and its

fourth,

fifth, and

sixth Fourier approximants.

Graph of

*f(x)* and its

seventh,

eighth, and

ninth Fourier approximants.

Graph of

*f(x)* and its

tenth,

fifteenth, and

twentieth Fourier approximants. (Note: To the resolution of this image,

*f(x)* and its twentieth Fourier approximant are indistinguishable.)

Here is a table of the first twenty Fourier coefficients of

*f(x)*.

*k* | *b*_{k} | *k* | *b*_{k} | *k* | *b*_{k} | *k* | *b*_{k} |

1 | 1.35174739 | 6 | .96466333 | 11 | -.22078270 | 16 | .07686692 |

2 | 3.17297840 | 7 | -.69694090 | 12 | .17369428 | 17 | -.06452967 |

3 | .51510823 | 8 | .50781673 | 13 | -.13886072 | 18 | .05467645 |

4 | 1.38915849 | 9 | -.37701467 | 14 | .11261631 | 19 | -.04671719 |

5 | -1.29334986 | 10 | .28569984 | 15 | -.09250886 | 20 | .04022103 |

*- Steven Weintraub*