# MATH310S26 - Day 12: From Fourier Series to Fourier Transform ## Course Information **Course**: MATH310 - Introduction to Mathematical Modeling **Date**: February 11, 2026 (Day 12) **Topics**: Fourier transform derivation, modes and nodes visualization, Dirac delta function **Previous Day**: [[MATH310S26-Day11-Notes]] ## Administrative Notes - MR02 feedback has been posted for all submissions - Next work day: Friday, February 13 - Options for Friday: - Code snippets for de-seasonalizing GMSL data - Individual project discussions - Weekly modeling reflections continue ## Main Topics ### 1. Motivation: Frequency Detection from Data **Driving Question**: If we get time series data, can we determine relevant frequencies from the data itself? Building on [[MATH310S26-Day11-Notes|Day 11's sea level analysis]], where we manually chose annual and semi-annual frequencies: - Previously: Forced projection onto year-long and twice-yearly frequencies - Goal: Infer frequencies directly from data - Solution: Generalize Fourier series → Fourier transform ### 2. Complex Fourier Series Review From [[MATH310S26-Day10-Notes|Day 10's complex form]]: $f(t) = \sum_{n=-\infty}^{\infty} c_n e^{i\omega_n t}$ where: - $\omega_n = \frac{n\pi}{L}$ (discrete frequencies) - $c(\omega_n) = \frac{1}{2L} \int_{-L}^{L} f(t) e^{-i\omega_n t} dt$ **Key Properties**: 1. **Periodicity**: $f(t + 2L) = f(t)$ due to $e^{2\pi i n} = 1$ for integer $n$ 2. **Complex conjugates**: For real $f(t)$, we need $c_{-n} = \overline{c_n}$ 3. **Real from complex**: $z + \overline{z} = 2\text{Re}(z)$ ensures real output ### 3. Geometric Interpretation of Complex Exponentials **Complex plane visualization**: - $e^{i\theta} = \cos(\theta) + i\sin(\theta)$ - Real part (horizontal): $\cos(\theta)$ - Imaginary part (vertical): $\sin(\theta)$ - **Result**: Unit circle in complex plane ### 4. Modes, Nodes, and Physical Vibrations #### Conceptual Hierarchy Wave → Shape → Mode → Nodes #### Visual Demonstrations **1D String Vibrations**: - [Fundamental frequency](https://en.wikipedia.org/wiki/Fundamental_frequency): No internal nodes - [Harmonics](https://en.wikipedia.org/wiki/Harmonic): $n$-th mode has $n-1$ internal nodes - Physical demonstration: Finger placement suppresses specific modes - See: [Dan Russell - The Plucked Fixed-Fixed String](https://www.acs.psu.edu/drussell/Demos/Pluck-Fourier/Pluck-Fourier.html) ![[Teaching/MATH310/MATH310S26/Lecture Notes/Media/string-standing-2.gif]] **2D Membrane Vibrations** ([Chladni patterns](https://en.wikipedia.org/wiki/Chladni_figures)): - Rectangular membranes: Lines and crosses as nodal patterns - Circular membranes: Radial and circular nodes - Sand settles at nodes (stationary points) - Applications: Musical instruments, architectural acoustics - See: [Dan Russell - Vibrational Modeshapes of a Rectangular Membrane (fixed at the edges)](https://www.acs.psu.edu/drussell/Demos/rect-membrane/rect-mem.html) ![[rect_24.gif]] * See: [Dan Russell - Vibrational Modes of a Circular Membrane](https://www.acs.psu.edu/drussell/Demos/MembraneCircle/Circle.html) ![[mode02-2.gif]] **Music and Modes**  **3D Droplet Oscillations**: - Water droplet suspended in acoustic field - Spherical harmonics visible as shape deformations - Higher modes → more complex patterns → eventual breakup  **Connection to Chemistry**: [Atomic orbitals](https://en.wikipedia.org/wiki/Atomic_orbital) as standing wave patterns ### 5. Breaking Periodicity: The Path to Fourier Transform #### The Limiting Process **Problem**: Fourier series requires periodicity with discrete frequencies $\omega_n = \frac{n\pi}{L}$ **Solution Strategy**: 1. Start with function on $[-L, L]$ 2. Let $L \to \infty$ to prevent periodic extension 3. Frequency spacing: $\Delta\omega = \frac{\pi}{L} \to 0$ 4. Discrete sum → continuous integral #### Mathematical Development Starting from the complex Fourier series: $f_L(t) = \sum_{n=-\infty}^{\infty} \left[\frac{1}{2L} \int_{-L}^{L} f_L(v) e^{-i\omega_n v} dv\right] e^{i\omega_n t}$ **Key substitutions**: - $\frac{1}{L} = \frac{\Delta\omega}{\pi}$ - Factor out $\frac{1}{2\pi}$ symmetrically **Rewrite as**: $f_L(t) = \frac{1}{2\pi} \sum_{n=-\infty}^{\infty} \left[\frac{1}{\sqrt{2\pi}} \int_{-L}^{L} f_L(v) e^{-i\omega_n v} dv\right] e^{i\omega_n t} \Delta\omega$ **Recognition**: This is a [Riemann sum](https://en.wikipedia.org/wiki/Riemann_sum)! - Heights: Bracketed expression (function of $\omega_n$) - Widths: $\Delta\omega \to d\omega$ - Sum over $n$ → integral over $\omega$ ### 6. The Fourier Transform Pair #### Forward Transform $\hat{f}(\omega) = \mathcal{F}\{f\}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$ **Interpretation**: - Input: Time domain signal $f(t)$ - Output: Frequency domain spectrum $\hat{f}(\omega)$ - Process: Projects onto continuous frequencies #### Inverse Transform $f(t) = \mathcal{F}^{-1}\{\hat{f}\}(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \hat{f}(\omega) e^{i\omega t} d\omega$ **Interpretation**: - Input: Frequency spectrum $\hat{f}(\omega)$ - Output: Time signal $f(t)$ - Process: Synthesizes from frequency components **Notation conventions**: - $\hat{f}$ or $F(\omega)$: Fourier transform of $f$ - $\mathcal{F}$: Fourier transform operator - $\mathcal{F}^{-1}$: Inverse Fourier transform operator ### 7. Mathematical Requirements and Historical Notes #### Function Space Considerations **Required**: [Absolute integrability](https://en.wikipedia.org/wiki/Absolute_convergence) $\int_{-\infty}^{\infty} |f(t)| dt < \infty$ **Why it works**: $|e^{-i\omega t}| = 1$ provides bounded scaling **20th Century Achievement**: Extension to [L² space](https://en.wikipedia.org/wiki/Square-integrable_function) (square-integrable functions) - [Lars Hörmander](https://en.wikipedia.org/wiki/Lars_H%C3%B6rmander) has one of the few texts I've found that has a complete rigorous treatments ([The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis (Classics in Mathematics)](https://www.amazon.com/Analysis-Linear-Partial-Differential-Operators/dp/3540006621/ref=sr_1_3?dib=eyJ2IjoiMSJ9.9a5RgFS_MdiU5zH6pYT_TlvYFxkGP_piCK83_Q2mrXtbA6OX92ee-jE1WCZkQjIO7Oi0kT3rMjRKrGFJdESCaUPeUEDsT1miPzcKa3OudaVDSRphysowQxNE-Y7SbYQXUyhsuH7J6XsES_onMX4r_USxXy1y4HLW_NYZXSs1uC-8F5DL1zR3jM1TUvg__Z44GYeb60Oq6ZWc7c7J3MqqkpbrZKSWr0ykFnei6v0Yeag.YOCCtOXArymHYNmKf3HTfIykQZ4gkNJW2RPRKsrVeXc&dib_tag=se&qid=1770850628&refinements=p_27%3ALars+Hormander&s=books&sr=1-3)) - Connects to [[MATH310S26-Day11-Notes|Day 11's inner product structure]] ### 8. Introduction to Dirac Delta Function #### Example Problem **Find the Fourier transform of** $f(t) = \cos(t)$ **Symmetry analysis**: - Cosine is even → transform should be even - Only frequencies present: $\omega = \pm 1$ **Result using [Dirac delta](https://en.wikipedia.org/wiki/Dirac_delta_function)**: $\hat{f}(\omega) = \sqrt{\frac{\pi}{2}}[\delta(\omega - 1) + \delta(\omega + 1)]$ **Properties of $\delta$**: - Zero everywhere except at origin - "Infinite" at origin (properly: distribution) - Unit area under "curve" **Connection**: Links continuous transforms back to discrete Fourier series ## Check Your Understanding 1. **Euler's identity proof**: Show $e^{i\theta} = \cos(\theta) + i\sin(\theta)$ using [Taylor series](https://en.wikipedia.org/wiki/Taylor_series) 2. **Complex Fourier verification**: Verify $c_n$ for $f(t) = 2$ (constant function) 3. **Delta function transform**: Show that $\mathcal{F}\{\cos(t)\}$ gives delta functions at $\omega = \pm 1$ (not included) ## Key Takeaways 1. **Fourier transform generalizes Fourier series** by removing periodicity constraint 2. **Continuous frequency spectrum** allows analysis of non-periodic signals 3. **Modes and nodes** provide physical intuition for abstract mathematics 4. **Complex exponentials** unify trigonometric functions geometrically 5. **Delta functions** bridge discrete and continuous frequency domains ## Mathematical Connections ### Previous Topics - [[MATH310S26-Day3-WorkdayMaterials (Two-by-two matrices, normal equations, and ordinary least squares)|OLS]] (Days 1-3): Projection onto basis functions - [[MATH310S26-Day5-Notes|Eigenanalysis]] (Days 4-6): Orthogonal decomposition - [[MATH310S26-Day7-Notes|Fourier series introduction]] (Day 7): Periodic functions - [[MATH310S26-Day10-Notes|Complex Fourier series]] (Day 10): Exponential form - [[MATH310S26-Day11-Notes|Sea level analysis]] (Day 11): Practical application ### Looking Ahead - [Fast Fourier Transform](https://en.wikipedia.org/wiki/Fast_Fourier_transform) (FFT): Computational algorithms - [Signal processing](https://en.wikipedia.org/wiki/Signal_processing): Filtering and analysis - [Spectral analysis](https://en.wikipedia.org/wiki/Spectral_analysis): Identifying system frequencies - Time-frequency analysis: [Wavelets](https://en.wikipedia.org/wiki/Wavelet) and [STFT](https://en.wikipedia.org/wiki/Short-time_Fourier_transform) ## References - [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform) - [Dirac delta function](https://en.wikipedia.org/wiki/Dirac_delta_function) - [Chladni figures](https://en.wikipedia.org/wiki/Chladni_figures) - [Riemann integral](https://en.wikipedia.org/wiki/Riemann_integral) - [Hilbert space](https://en.wikipedia.org/wiki/Hilbert_space) - [Distribution theory](https://en.wikipedia.org/wiki/Distribution_(mathematics))