Harmonic Analysis | Research & Encyclopedia Articles

Harmonic Analysis

Harmonic analysis is the branch of mathematics which developed from the study of Fourier series, Fourier transforms, and other related operators. One of the most basic problems in harmonic analysis is to determine how periodic functions can be written as a sum of trigonometric functions. Suppose, for example, that ƒ(x) is a real or complex valued function defined for all real numbers x, and ƒ is also periodic with period 1. By periodic with period 1 we mean that ƒ satisfies the identity ƒ(x+1) = ƒ(x) for all real numbers x. It follows then that ƒ(x+n) = ƒ(x) for all integers n. By considering a graph of y = ƒ(x) it is apparent that ƒ is completely determined by its behavior on any interval of length 1, for example, the interval [0,1]. Now some of the simplest examples of periodic functions with period 1 are the trigonometric functions

cos 2x, cos 4x, cos 6x, ... , cos 2lx, ... ,

where l is a positive integer, and

sin 2x, sin 4x, sin 6x, ... , sin 2mx, ... ,

where m is a positive integer. The constant function 1 is also periodic with period 1 and it is convenient to include this as cos 2lx with l = 0. If we form a linear combination of these functions, such as

where a₀, a₁, a₂, ... , a_L and b₁, b₂, b₃, ... , b_M are real or complex coefficients, then we continue to get a function which is periodic with period 1. Now it is natural to ask if more general functions ƒ(x), which are periodic with period 1, can be written as sums of this sort. Since we want to reach as many functions ƒ(x) as possible, we should also allow infinite series expansions rather than just finite sums.

In order to describe this problem and its solution more easily, it is a good idea to replace the functions cos 2lx and sin 2mx with the complex exponential functions

e^2inx = cos 2nx + isin 2nx where n = ... -3, -2, -1, 0, 1, 2, 3,....

These functions are also periodic with period 1 and allow us to work with cos and sin simultaneously. Now the basic problem can be restated as follows: if ƒ(x) is periodic with period 1, does there exist a sequence of real or complex numbers ... c_-2, c_-1, c₀, c₁, c₂, ... such that

And if these numbers exist, then how can we compute them from our knowledge of ƒ? Here is one solution: assume that ƒ is a continuous function and then define the complex number c_n by the formula

Also, assume that

that is, the infinite series formed with the numbers c_n is absolutely convergent. Under these assumptions we have

at each real number x. The coefficients c_n are called the Fourier coefficients of ƒ. It is obvious that they are determined by the function ƒ(x) and to indicate this they are often written (n). Notice that we may regard : Z C as a function from the integers to the complex numbers that has been determined by the original periodic function ƒ(x).

Here is a second solution: assume that ƒ is a continuous function and let the complex numbers c_n = (n) be defined as before. In this case we make no assumption about the absolute convergence of the infinite series formed with the numbers (n). Then we have

at each real number x. Notice that in this second solution we have introduced the additional factors into the sum. Some modification of this sort is necessary because of the following surprising fact. There exists a continuous function ƒ(x) and a real number ξ such that

Here is an example: let g(x) = |sin x|. Then it is easy to check that g is periodic with period 1 and that g is continuous. The Fourier coefficients of g are given for each integer n as the value of the integral

Since the infinite series

is convergent, we have

at each real number x.

The process of going from the periodic function ƒ(x) to the function : Z C is an example of the Fourier transform. Then a basic problem of harmonic analysis is to learn how to recover the function ƒ(x) from knowledge of . This process can also be formulated in other settings. Suppose that ƒ: ℜ C$ is a Lebesgue integrable function, (see the article on the Lebesgue integral). In this case we define the Fourier transform of ƒ to be the function : ℜ C defined by the Lebesgue integral

Although we have continued to use the same notation as for periodic functions and their Fourier coefficients, in fact the situation now is quite different. The function ƒ(x) is defined for real numbers x and is not assumed to be periodic, and the function (t) is also defined for real numbers t. Because ƒ is an integrable function, it can be shown that (t) is a continuous function and also satisfies

In this setting we may also ask how the function ƒ(x) can be recovered from its Fourier transform (t). We can give a reasonably simple answer by imposing two additional conditions which in practice are often satisfied. Assume that ƒ(x) is continuous and that (t) in integrable on ℜ. Then with these assumptions we have

This is called the Fourier inversion formula. Notice that the inversion formula is nearly the same as the formula defining the Fourier transform.

Here is an example: let h(x) = e^-2|x|. Then the Fourier transform of h is the function ĥ: ℜ C defined for each real number t by

Because h is continuous and ĥ is integrable, the Fourier inversion formula provides the identity

for all real numbers x.

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