Unit 3: Fourier Series on Bounded Domains As you may have learned in a physics (or music!) class, every sound is made up of sound waves with different frequencies. Thus, for instance, a C-major chord based at middle C is made of up waves at 261.6 Hz(C), 329.6 Hz (E), and 392.0 Hz (G). The combination (or superposition) of these three individual sound waves produces a new and distinct sound. In fact, any signal or waveform (subject to some reasonable conditions) can be written as the combination of some fundamental waves, whether it is a water wave, a light or electromagnetic wave, or a sound wave. Mathematically, a waveform is just a function, and, conversely, every function can be viewed as a waveform. Most (again, subject to some reasonable conditions) can be written as a combination of some fundamental wave-like functions, namely sines and cosines. This was the observation of Fourier, a great scientist of the nineteenth century. It took years for other mathematicians to accept his work and come up with a theoretical framework for his results. The study of such ideas is today called Harmonic Analysis, and it has gone far beyond Fourier’s original ideas to touch almost every part of modern science and technology, from medical imaging to wireless communication.
In this unit, you will learn to apply Fourier’s original insight – that most useful functions can be written as linear combinations of sine and cosine waves. To do this rigorously, some mathematical underpinnings must be established, such as the definition and elaboration of the square-integrable functions, the differences between different modes of convergence, and the orthogonality of the basis of trigonometric functions, some of which you will recall from MA241. After this has been accomplished, you will learn how to calculate Fourier series for suitable functions.
Unit 3 Time Advisory
This unit should take you approximately 14.5 hours to complete.
☐ Subunit 3.1: 4.75 hours Subunit 3.1.1: 0.75 hours ☐
☐ Subunit 3.1.2: 0.25 hours
☐ Subunit 3.1.3: 0.5 hours
☐ Subunit 3.1.4: 3 hours
☐ Subunit 3.1.5: 0.25 hours
☐ Subunit 3.2: 9.75 hours ☐ Subunit 3.2.1: 4 hours
☐ Subunit 3.2.2: 1 hour
☐ Subunit 3.2.3: 0.25 hours
☐ Subunit 3.2.4: 4.5 hours
Unit3 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- define, characterize, and use inner products;
- define the space of L^{2} functions and state its key
properties;
- show that a function belongs to the space of
L^{2 }functions;
- define orthogonality and show the orthogonality of certain
trigonometric functions;
- define the Haar Basis;
- distinguish between pointwise, uniform, and L^{2}
convergence;
- show convergence of sequences;
- use the Weierstrass M-test to show that Fourier series converge;
- define orthogonal and orthonormal basis;
- define Fourier series on [0,π] and [0,L] and identify sufficient
conditions for their convergence and uniqueness;
- define the Gibbs Phenomenon; and
- compute Fourier coefficients and construct Fourier series.
3.1 Necessary Functional Analysis
3.1.1 Inner Products
- Reading: Cambridge University Press: Professor Marcus Pivato’s
Linear Partial Differential Equations and Fourier Theory: “6A:
Some Functional Analysis: Inner Products”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear
Partial Differential Equations and Fourier Theory: “6A: Some
Functional Analysis: Inner
Products”
(PDF)
Instructions: Click on the link above and read section 6A (pages
103-105).
Solving linear PDEs using algebraic methods requires an
understanding of inner product spaces.
Studying this reading should take approximately 20 minutes.
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- Assessment: MIT: Professor Steven Johnson’s “Problem Set 2”
Link: MIT: Professor Steven Johnson’s “Problem Set
2”
(PDF)
Instructions: Click on the link above to access the PDF file, and complete problem 2 from “Problem Set 2.” To check your solutions, follow this link.
You should dedicate approximately 30 minutes to completing this assessment.
Terms of Use: The materials above are released under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States (CC BY-NC-SA 3.0) (HTML).
3.1.2 L2 Space
- Reading: Cambridge University Press: Professor Marcus Pivato’s
Linear Partial Differential Equations and Fourier Theory: “6B:
Some Functional Analysis: L2 Space”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear
Partial Differential Equations and Fourier Theory: “6B: Some
Functional Analysis: L2
Space”
(PDF)
Instructions: Click on the link above and read section 6B (pages
105-108).
In order to apply ideas from linear algebra to PDEs, one requires a
normed linear space with a well-defined norm. Using the inner
product ideas from the previous section, the author defines the
vector space L^{2} of square-integrable functions.
Studying this reading should take approximately 15-20 minutes.
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3.1.3 Orthogonality
- Reading: Cambridge University Press: Professor Marcus Pivato’s
Linear Partial Differential Equations and Fourier Theory: “6D:
Some Functional Analysis: Orthogonality”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear
Partial Differential Equations and Fourier Theory: “6D: Some
Functional Analysis:
Orthogonality”
(PDF)
Instructions: Click on the link above and read section 6D (pages
112-116).
In this section, the author shows the orthogonality of
trigonometric functions, which will be a crucial fact for the
development of Fourier series in upcoming units.
Studying this reading should take approximately 30 minutes.
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3.1.4 Convergence Concepts
- Reading: Cambridge University Press: Professor Marcus Pivato’s
Linear Partial Differential Equations and Fourier Theory: “6E:
Some Functional Analysis: Convergence Concepts”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear
Partial Differential Equations and Fourier Theory: “6E: Some
Functional Analysis: Convergence
Concepts”
(PDF)
Instructions: Click on the link above and read section
6E (pages 116-131).
In many cases, solutions to PDEs can be constructed using sequences
or series of functions. However, as you know from calculus, the
construction of a sequence or series does not guarantee that it has
a well-defined limit. For this reason, the author defines several
modes of convergence, which will be used to establish the validity
of these types of solutions. The diagrams used to illustrate these
concepts are excellent.
Studying this reading should take approximately 3 hours.
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3.1.5 Orthogonal and Orthonormal Bases
- Reading: Cambridge University Press: Professor Marcus Pivato’s
Linear Partial Differential Equations and Fourier Theory: “6F:
Some Functional Analysis: Orthogonal and Orthonormal Bases”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear
Partial Differential Equations and Fourier Theory: “6F: Some
Functional Analysis: Orthogonal and Orthonormal
Bases”
(PDF)
Instructions: Click on the link above and read section
6F (pages 131-133).
Studying this reading should take approximately 15-20 minutes.
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3.2 Fourier Series
3.2.1 Eigensolutions to Linear Evolution Equations
- Reading: University of Minnesota: Professor Peter Olver’s
Introduction to Partial Differential Equations: “Chapter 3:
Fourier Series: 3.1: Eigensolutions to Linear Evolution Equations”
Link: University of Minnesota: Professor Peter Olver’s Introduction
to Partial Differential Equations: “Chapter 3: Fourier Series:
3.1: Eigensolutions to Linear Evolution
Equations”
(PDF)
Instructions: Click on the link above. Scroll down to the Chapter
3, and click on the link to download the PDF. Read pages 55-63.
In this reading, the author motivates the construction of Fourier
series by tying them to eigenvalue problems for linear evolution
equations. We will return to these ideas when we discuss the
technique of separation of variables.
Studying this reading should take approximately 4 hours.
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3.2.2 Fourier Series on [0, ?] - Reading: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “7A: Fourier Sine Series and Cosine Series: Fourier (Co)sine Series on [0, ?]” Link: Cambridge University Press: Professor Marcus Pivato’s Linear Partial Differential Equations and Fourier Theory: “7A: Fourier Sine Series and Cosine Series: Fourier (Co)sine Series on [0, π]” (PDF)
Instructions: Click on the link above and read section 7A (pages
137-144).
All of the facts that were established in section 3.1 about inner
products, orthogonality, the L<sup>2</sup> functions, and
orthonormal bases are brought into play in this section. Fourier
series are one of your most powerful tools for solving PDEs. They
are also extremely important for signal and image processing, so
investment in understanding this concept is well-worth your time.
Studying this reading should take approximately 1 hour.
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3.2.3 Fourier Series on [0,L]
- Reading: Cambridge University Press: Professor Marcus Pivato’s
Linear Partial Differential Equations and Fourier Theory: “7B:
Fourier Sine Series and Cosine Series: Fourier (Co)sine Series on
[0, L]”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear
Partial Differential Equations and Fourier Theory: “7B: Fourier
Sine Series and Cosine Series: Fourier (Co)sine Series on [0,
L]”
(PDF)
Instructions: Click on the link above and read section 7B (pages
144-146).
Studying this reading should take approximately 15-20 minutes.
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displayed on the webpages above.
3.2.4 Computing Fourier Coefficients
- Reading: Cambridge University Press: Professor Marcus Pivato’s
Linear Partial Differential Equations and Fourier Theory: “7C:
Fourier Sine Series and Cosine Series: Computing Fourier (Co)sine
Coefficients”
Link: Cambridge University Press: Professor Marcus Pivato’s Linear
Partial Differential Equations and Fourier Theory: “7C: Fourier
Sine Series and Cosine Series: Fourier (Co)sine
Coefficients”
(PDF)
Instructions: Click on the link above and read section 7C (pages
147-158).
This section reviews some important methods of computing Fourier
coefficients for various functions, including polynomials, step
functions, and piecewise linear functions.
Studying this reading should take approximately 1 hour and 30
minutes.
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Assessment: MIT: Professor Matthew Hancock’s “Problem Set 1” and Trinity College, Dublin: Professor Sarah McMurry’s “Problem Sheet 2” Link: MIT: Professor Matthew Hancock’s “Problem Set 1” (PDF) and Trinity College, Dublin: Professor Sarah McMurry’s “Problem Sheet 2” (PDF)
Instructions: Click on the first link above to access the PDF, and complete problem 1 from “Problem Set 1.” Then, click on the second link above, scroll down to the section “Problem Sheets,” and select the link to “Problem Sheet 4.” Complete all of the problems on “Problem Sheet 4.” To check your solutions for the Hancock assessments click on this link. To check your solutions for the McMurry assessments follow the second link again and click on the corresponding link (“Answers to Sheet 4”) for solutions.
You should dedicate approximately 3 hours to completing this assessment.
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