# Why are Fourier series important?

• matqkks
In summary, the conversation discusses the real life applications of Fourier series and the search for examples that can motivate students to study this topic. Some examples mentioned include signal processing, electrical principles, laser science, sound processing, and solving linear differential equations in physics and engineering. The conversation also highlights the importance of understanding linear algebra in comprehending Fourier series. The original historical motivation for Fourier series, such as heat transfer and the vibrating string problem, are also mentioned as practical applications. However, the practicality of these specific problems is debated.
matqkks
Are there any real life applications of Fourier series?

Are there examples of Fourier series which have an impact on students learning this topic. I have found the normal suspects of examples in this field such as signal processing, electrical principles but there must be a vast range of applications as it is taught on most undergraduate courses in mathematics, physics and engineering. I am looking for examples which will motivate why students should study Fourier series.

Sounds produced by musical instruments.

matqkks said:
Are there any real life applications of Fourier series?

I am looking for examples which will motivate why students should study Fourier series.

The trouble with motivating mathematics by showing applications is that you have teach the details of the applications to convey the full meaning. You can always give a little speech like "Fourier series are useful in electronics, acoustics, ...blah-blah" and not show the details, but I don't think that's what you have in mind.

We can consider whether there are any gizmos you could set up on a table top and show how sums of pure trigonometric functions add to a complicated function - and the reverse process. I don't know of any simple equipment that does this. You can do it with electronic devices, but it isn't convincing unless the class is already familiar with the equipment.

I think such a demonstration is actually misleading. In applications of Fourier series, it is usually implausible to think that a complicated function is actually being produced by some physical gizmos that are producing pure sine or cosine waves. It is analogous to the case with vectors. We can imagine a vector being the result of two component vectors in various ways, but in a physical situation there is often no physical implementation of each separate component..

I think Fourier series are easier to swallow if the students first understand the general idea of linear independence and orthogonality as applied to functions. If we are given a set of simple looking functions and a complicated function there are two basic questions 1) Can we approximate the complicated function using the simpler functions? 2) How do we find the approximation? Question 1) is mathematically complicated. Question 2) can be taught by analogy to vectors. If you have a non-independent set of non-orthogonal vectors, you may be able to represent a given vector in various non-unique ways. If you want to represent in a convenient and unique way, you should use a set of independent vectors, if possible a set of orthogonal vectors, and even better a set of orthonormal vectors.

it might also help if they know about eigenvectors.

matqkks said:
Are there any real life applications of Fourier series?

Are there examples of Fourier series which have an impact on students learning this topic. I have found the normal suspects of examples in this field such as signal processing, electrical principles but there must be a vast range of applications as it is taught on most undergraduate courses in mathematics, physics and engineering. I am looking for examples which will motivate why students should study Fourier series.

It is important in so many ways in laser science. (mode locking, bandwidth versus pulse length, etc.)

Fourier series aren't used so much, but the Fourier transform is related and is totally ubiquitous in its applications. The Fourier series is a good stepping stone toward the Fourier transform.

Sound processing is probably the purest and simplest application.

Fourier series are important for understanding Fourier Transforms which is one of the most basic elements of signal processing of all sorts (including Khashishi's sound processing). If you are interested in that subject, a good book is the University of Lex's "Who is Fourier. Just be warned that even though the book looks like (and is) a cartoon book, the math in it is difficult and complex for those with less than a couple years of college math.

Many of the basic areas of physics and engineering can be mathematically described by differential equations - heat flow, fluid dynamics, electromagnetic theory, quantum mechanics. Fourier series naturally arise in solving some linear differential equations (heat equation, wave equation, Schrodinger's equation, etc.) in bounded regions. I say "naturally" because they are eigenfunctions of a differential operator (see Mathwonk's comment above). This application alone would be motivation for including it in basic mathematical training of physicists and engineers.

Today, I would think the discrete version is at least as important, as much of the computation done today is in the digital domain. Discrete time Fourier series are essentially the discrete Fourier transform (DFT). And in the digital domain, if we have a short signal that is not periodic, we often pretend that the entire signal is one period of a periodic signal and use the DFT to analyze/process it. There are a fast set of algorithms collectively known as Fast Fourier Transforms (FFT) that efficiently calculate the DFT. The FFT is so efficient that engineers will often go out of their way to be able to use the FFT instead of something much more computationally expensive. I think it is fair to say that the FFT has had a large impact on computation for many applications, and the FFT is simply a fast way of computing the coefficients of a discrete time Fourier series.

Fourier analysis is one of the most important tools that I use as an electrical engineer. As Stephen Tashi and Mathwonk indicated, knowing linear algebra really helps with understanding Fourier series.

jason

Why not use Fourier's reason? Heat transfer. Cooling off a 1-dimensional rod. Similarly, the vibrating string problem, which was the original historical motivation. I don't know that these specific problems are that important, in terms of really practical applications, but they do correspond to actual physics experiments that you can carry out.

To me the best motivation is to note that you can get any twice differentiable function to satisfy the wave equation if you move it along at the right velocity, but on the other hand, if you know that you can think of the vibrating string as a limiting case of a bunch of harmonic oscillators, and you know about normal modes in physics, it should be the case that any motion of the string is a superposition of the normal modes (i.e. the eigenfunctions). If you put the two points of view together, it seems physically plausible that you ought to be able to represent functions in terms of the normal modes. Of course, this sort of depends on understanding normal modes, which could be a big digression, depending on the students' background, but with some finesse, you could probably develop this explanation to be accessible to audiences at different levels, with correspondingly different levels of hand-waving.

In the case of the cooling off problem for a rod, we can work out that sinusoidal things decay exponentially. So, if, as we've been taught to do from studying the vibrating string, we can express the initial temperature distribution as a Fourier series, we should be able to solve the problem for a general initial temperature distribution, since we know how to solve it for sinusoidal ones.

## 1. Why do we use Fourier series to represent functions?

Fourier series are used to represent functions because they provide a way to break down a complex function into simpler components. These simpler components are sinusoidal functions that are easier to manipulate and analyze mathematically.

## 2. How are Fourier series used in practical applications?

Fourier series are used in a wide range of practical applications, including signal processing, image and sound compression, and data analysis. They are also used in fields such as physics, engineering, and mathematics to model and analyze various phenomena.

## 3. What is the significance of the Fourier series coefficients?

The Fourier series coefficients represent the amplitudes and phases of the sinusoidal components that make up a function. They provide important information about the behavior and characteristics of the function, such as its periodicity and symmetry.

## 4. Can all functions be represented by a Fourier series?

No, not all functions can be represented by a Fourier series. The function must be periodic and have a finite number of discontinuities to be represented accurately. However, many functions can be approximated by a Fourier series with a sufficient number of terms.

## 5. How do Fourier series relate to the concept of harmonics?

Fourier series are closely related to the concept of harmonics, which refers to the sinusoidal components that make up a complex wave. The terms in a Fourier series can be thought of as different harmonics with different frequencies and amplitudes. The more terms included in the series, the closer the approximation to the original function, similar to how adding more harmonics can more accurately reproduce a complex sound wave.

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