Standard syllabus
Fourier analysis & transforms · Undergraduate · Math
Learning objectives from the Fourier analysis & transforms syllabus, grouped by unit. Click an objective for study materials.
Topics typically covered
Click a topic for the full text and related unit practice.
Undergraduate Fourier Analysis — scope drawn from open Fourier analysis / transforms notes and typical US applied Fourier syllabi.
Fourier series
- Orthogonality of sine — Orthogonality of sine
- Cosine functions on intervals — Cosine functions on intervals
- Fourier series for periodic functions — Fourier series for periodic functions; coefficients and convergence
- Even and odd extensions — Even and odd extensions; sine and cosine series
- Parseval's theorem — Parseval's theorem
- Energy interpretation — Energy interpretation
- Gibbs phenomenon — Gibbs phenomenon
- Convergence at discontinuities — Convergence at discontinuities
Fourier and Laplace transforms
- Definition — Definition
- Properties of the Fourier transform — Properties of the Fourier transform
- Convolution theorem and filtering — Convolution theorem and filtering
- Discrete Fourier transform (DFT) and sampling — Discrete Fourier transform (DFT) and sampling
- Laplace transform review — Laplace transform review
- Connection to frequency domain — Connection to frequency domain
- Inverse transforms — Inverse transforms
- Transform pairs (table methods) — Transform pairs (table methods)
Applications and generalizations
- Heat equation and PDE solutions via Fourier series — Heat equation and PDE solutions via Fourier series
- Signal decomposition — Signal decomposition
- Frequency content — Frequency content
- Window functions — Window functions
- Leakage (introduction) — Leakage (introduction)
- Fast Fourier transform (FFT) algorithm overview — Fast Fourier transform (FFT) algorithm overview
- Introduction to wavelets (optional topic) — Introduction to wavelets (optional topic)
Learning objectives
Click an objective for study materials.
Fourier series
- Orthogonality of sine — Orthogonality of sine
- Cosine functions on intervals — Cosine functions on intervals
- Fourier series for periodic functions — Fourier series for periodic functions; coefficients and convergence
- Even and odd extensions — Even and odd extensions; sine and cosine series
- Parseval's theorem — Parseval's theorem
- Energy interpretation — Energy interpretation
- Gibbs phenomenon — Gibbs phenomenon
- Convergence at discontinuities — Convergence at discontinuities
Fourier and Laplace transforms
- Definition — Definition
- Properties of the Fourier transform — Properties of the Fourier transform
- Convolution theorem and filtering — Convolution theorem and filtering
- Discrete Fourier transform (DFT) and sampling — Discrete Fourier transform (DFT) and sampling
- Laplace transform review — Laplace transform review
- Connection to frequency domain — Connection to frequency domain
- Inverse transforms — Inverse transforms
- Transform pairs (table methods) — Transform pairs (table methods)
Applications and generalizations
- Heat equation and PDE solutions via Fourier series — Heat equation and PDE solutions via Fourier series
- Signal decomposition — Signal decomposition
- Frequency content — Frequency content
- Window functions — Window functions
- Leakage (introduction) — Leakage (introduction)
- Fast Fourier transform (FFT) algorithm overview — Fast Fourier transform (FFT) algorithm overview
- Introduction to wavelets (optional topic) — Introduction to wavelets (optional topic)
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Fourier series
Orthogonality of sine
Coming soon - Fourier and Laplace transforms
Definition
Coming soon - Applications and generalizations
Heat equation and PDE solutions via Fourier series
Coming soon
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$1,162 · Fourier analysis & transforms · 18 tutoring hrs
Study guides, worksheets, reviews, practice tests, and answer keys for 1 class. 18 tutoring hours (1 hr / week · semester). Bundle discount applied vs buying separately. Pay in full via Zelle.