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MA304: Topics in Applied Mathematics

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.
 
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  • 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.
     
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3.1.1 Notion of Fourier Series   - Reading: Notion of Fourier Series The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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3.1.2 Orthogonality and Computation   - Reading: Orthogonality and Computation The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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3.2 Orthogonal Projection   - Reading: Orthogonal Projection The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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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 The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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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 The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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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 The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

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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