ee261text

ee261text - Contents 1 Fourier Series 1 1.1 Introduction...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Contents 1 Fourier Series 1 1.1 Introduction and Choices to Make . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Periodic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Periodicity: Definitions, Examples, and Things to Come . . . . . . . . . . . . . . . . . . . . 4 1.4 It All Adds Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Lost at c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Period, Frequencies, and Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.7 Two Examples and a Warning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.8 The Math, the Majesty, the End . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.9 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.10 Appendix: The Cauchy-Schwarz Inequality and its Consequences . . . . . . . . . . . . . . . 33 1.11 Appendix: More on the Complex Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.12 Appendix: Best L 2 Approximation by Finite Fourier Series . . . . . . . . . . . . . . . . . . 38 1.13 Fourier Series in Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.14 Notes on Convergence of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1.15 Appendix: Pointwise Convergence vs. Uniform Convergence . . . . . . . . . . . . . . . . . . 59 1.16 Appendix: Studying Partial Sums via the Dirichlet Kernel: The Buzz Is Back . . . . . . . . 60 1.17 Appendix: The Complex Exponentials Are a Basis for L 2 ([0 , 1]) . . . . . . . . . . . . . . . . 62 1.18 Appendix: More on the Gibbs Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2 Fourier Transform 65 2.1 A First Look at the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.2 Getting to Know Your Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3 Convolution 95 3.1 A * is Born . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.2 What is Convolution, Really? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.3 Properties of Convolution: It’s a Lot like Multiplication . . . . . . . . . . . . . . . . . . . . 101 ii CONTENTS 3.4 Convolution in Action I: A Little Bit on Filtering . . . . . . . . . . . . . . . . . . . . . . . . 102 3.5 Convolution in Action II: Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.6 Convolution in Action III: The Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . 116 3.7 The Central Limit Theorem: The Bell Curve Tolls for Thee . . . . . . . . . . . . . . . . . . 128 3.8 Fourier transform formulas under different normalizations . . . . . . . . . . . . . . . . . . . ....
View Full Document

{[ snackBarMessage ]}

Page1 / 428

ee261text - Contents 1 Fourier Series 1 1.1 Introduction...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online