**Unit 3: Fourier Methods**
*The matrix transformation DFT introduced in Unit 2 is a discrete
version of the Fourier series to be studied in this unit. The theory of
Fourier series is very rich. For example, partial sums of the Fourier
series are orthogonal projection of the function it represents to the
corresponding subspaces of trigonometric polynomials. In addition,
these partial sums can be formulated as convolution of the function with
the “Dirichlet kernels.” Since averaging of the Dirichlet kernels
yields the “Fejer kernels” that constitute a positive “approximate
identity,” it follows that convergence, in the mean-square sense, of the
sequence of trigonometric polynomials, resulting from convolution of the
function with the Fejer kernels, to the function itself is assured.
Consequently, being orthogonal projections, the partial sums of the
Fourier series also converge to the function represented by the Fourier
series, again in the mean-square sense. This introduces the concept of
completeness, which is shown to be equivalent to Parseval’s identity,
with such interesting applications as solving the famous the Basel
problem. This unit explores examples of the extension of the original
Basel problem from powers of 2 to powers of 4 and to powers of 6. The
completeness property of Fourier series will be applied to solving
boundary value problems of PDE in Unit 5.*

**Unit3 Learning Outcomes**

Upon successful completion of this unit, the student will be able to:

- Compute Fourier coefficients.
- Compute Fourier cosine series.
- Compute mean-square error of approximation by partial sums of
Fourier series.
- Solve the Basel problem and its extension by using functions
different from the examples given in this unit.

**3.1 Fourier Series**
- **Reading: University of Minnesota: Professor Peter Olver’s
Introduction to Partial Differential Equations: “Chapter 3: Fourier
Series:” “Section 3.1: Eigensolutions to Linear Evolution
Equations”**
Link: University of Minnesota: Professor Peter Olver’s
*Introduction to Partial Differential
Equations:* “Chapter 3:
Fourier Series:” “Section 3.1: Eigensolutions to Linear Evolution
Equations” (PDF)

Instructions: Please click on the link above to access the table of
contents for the text, and select the link to download the PDF of
“Chapter 3: Fourier Series.” Study Section 3.1 on pages 63–71 to
learn about eigensolutions to linear evolution equations.

Studying this reading should take approximately 45 minutes to
complete.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

**Lecture: MIT: Professor Gilbert Strang’s Computational Science and Engineering I: “Lecture 28: Fourier Series (Part 1)” and “Lecture 29: Fourier Series (Part 2)”**Link: MIT: Professor Gilbert Strang’s Computational Science and Engineering I: “Lecture 28: Fourier Series (Part 1)” (YouTube) and “Lecture 29: Fourier Series (Part 2)” (YouTube)Instructions: Please click on the links above, and view this two-part lecture on Fourier series.

Viewing these lectures and pausing to take notes should take approximately 2 hours to complete.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages above.

**3.1.1 Notion of Fourier Series**
- **Reading: Notion of Fourier Series**
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**3.1.2 Orthogonality and Computation**
- **Reading: Orthogonality and Computation**
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**3.2 Orthogonal Projection**
- **Reading: Orthogonal Projection**
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**3.2.1 Pythagorean Theorem**
**3.2.2 Parallelogram Law**
**3.2.3 Best Mean-Square Approximation**
**3.3 Dirichlet’s and Fejer’s Kernels**
- **Reading: Dirichlet’s and Fejer’s Kernels**
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**3.3.1 Partial Sums as Convolution with Dirichlet’s Kernels**
**3.3.2 Cesaro Means and Derivation of Fejer’s Kernels**
**3.3.3 Positive Approximate Identity**
**3.4 Completeness**
- **Reading: Completeness**
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**3.4.1 Pointwise and Uniform Convergence**
**3.4.2 Trigonometric Approximation**
**3.5 Parseval’s Identity**
- **Reading: Parseval’s Identity**
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**3.5.1 Derivation**
**3.5.2 The Basel Problem and Fourier Method**
**3.5.3 Bernoulli Numbers and Euler’s Solution**