# MATH310S26 - Introduction to Mathematical Modeling (Lecture Notes) ## Spring 2026 - Day 7 Notes ### Date: January 28, 2026 (01/28/26) **Previous Lecture**: [[MATH310S26-Day6-Work|Day 6 - Work Day]] **Next Lecture**: Day 8 (upcoming) **Related Topics**: [[MATH310S26-Day1-Notes|Course Introduction]], [[MATH310S26-Day2-Notes|Linear Regression Basics]] --- ## Administrative Updates ### Modeling Survey - Posted and due Friday - Part of A-level and B-level long-term projects - Weekly check-ins to understand student interests and directions - Not expected to take much time - just to gauge where everyone's thinking ### Upcoming on Monday - Low-stakes feedback based on today's lecture and next lecture - Some coding exercises - **Group formation discussion** - Start thinking about working individually vs. in groups - Spring class (~20 students) allows flexibility - Fall class (~30 students) typically requires more group work ### Assignment Note - Some hyperlinks in CSV data assignment not working - Can navigate to website directly - Flexible on deadline if issues arise --- ## Main Topic: Introduction to Fourier Series ### Part 1: Building Functions from Periodic Components #### Recall: Regression Framework We can try to fit data to features through [regression](https://en.wikipedia.org/wiki/Regression_analysis). This connects to our earlier work with [[MATH310S26-Day2-Notes|linear regression]] and [[MATH310S26-Day3-WorkdayMaterials (Two-by-two matrices, normal equations, and ordinary least squares)|ordinary least squares]]. #### The Fundamental Set Consider the set of **2π-periodic [sinusoids](https://en.wikipedia.org/wiki/Sine_wave)**: $S = \{1, \cos(t), \sin(t), \cos(2t), \sin(2t), \cos(3t), \sin(3t), ...\}$ Key observations: - $\sin(t)$ has [period](https://en.wikipedia.org/wiki/Periodic_function) $2\pi$ - $\sin(2t)$ has period $\pi$, but also repeats every $2\pi$ - The constant function 1 is periodic with any period - All elements share a common period of $2\pi$ #### Linear Combinations The natural thing to do: **linearly combine the set's elements** $f(t) = a_0 + \sum_{n=1}^{\infty} [a_n \cos(nt) + b_n \sin(nt)]$ Where: - $a_0, a_n, b_n$ are regression parameters (coefficients) - $t$ is the independent variable (predictor) - $f(t)$ is the response ### Part 2: The Fourier Series #### Definition If we build smooth functions by linear combinations of: - **Power functions** → [Taylor Series](https://en.wikipedia.org/wiki/Taylor_series) - **Sinusoids** → **[Fourier Series](https://en.wikipedia.org/wiki/Fourier_series)** #### Key Properties **Theorem**: If $f_1$ and $f_2$ are periodic with period $p$, then $g(t) = f_1(t) + f_2(t)$ is also $p$-periodic. **Proof sketch**: $g(t+p) = f_1(t+p) + f_2(t+p) = f_1(t) + f_2(t) = g(t)$ **Consequence**: The Fourier series $f(t)$ is $2\pi$-periodic. #### Physical Interpretation - Think of sines and cosines as **[waves](https://en.wikipedia.org/wiki/Wave)** - Coefficients $a_n$ and $b_n$ control **[amplitudes](https://en.wikipedia.org/wiki/Amplitude)** - Linear combination creates **[superposition](https://en.wikipedia.org/wiki/Superposition_principle)** - Results in **[constructive and destructive interference](https://en.wikipedia.org/wiki/Wave_interference)** **Analogy**: A [prism](https://en.wikipedia.org/wiki/Prism_(optics)) breaking white light into rainbow colors is like a physical Fourier decomposition. ### Part 3: Finding Fourier Coefficients #### The Problem Given target data $f(t)$, how do we find the right $a_0, a_n, b_n$? **Challenge**: We need to find infinitely many coefficients! #### Example: Constant Function Suppose $f(t) = 2$ for $t \in (-\pi, \pi)$ and $f$ is $2\pi$-periodic. **Visual**: A horizontal line at $y=2$ that repeats every $2\pi$. #### Method 1: Finding $a_0$ **Step 1**: Integrate the entire Fourier series equation from $-\pi$ to $\pi$: $\int_{-\pi}^{\pi} f(t) dt = \int_{-\pi}^{\pi} \left[a_0 + \sum_{n=1}^{\infty} (a_n \cos(nt) + b_n \sin(nt))\right] dt$ **Step 2**: Apply linearity of integration (switching sum and integral): $\int_{-\pi}^{\pi} f(t) dt = a_0 \cdot 2\pi + \sum_{n=1}^{\infty} \left[a_n \int_{-\pi}^{\pi} \cos(nt) dt + b_n \int_{-\pi}^{\pi} \sin(nt) dt\right]$ **Key observation**: - $\int_{-\pi}^{\pi} \cos(nt) dt = 0$ (full periods integrate to zero) - $\int_{-\pi}^{\pi} \sin(nt) dt = 0$ (full periods integrate to zero) **Result**: $a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) dt$ For our example: $a_0 = \frac{1}{2\pi} \cdot 2 \cdot 2\pi = 2$ #### Method 2: Finding $a_n$ **Technique**: Multiply by $\cos(mt)$ and integrate from $-\pi$ to $\pi$ **[Orthogonality relations](https://en.wikipedia.org/wiki/Orthogonal_functions)** (Check Your Understanding): $\int_{-\pi}^{\pi} \cos(nt)\cos(mt) dt = \begin{cases} 0, & n \neq m \\ \pi, & n = m \end{cases}$ $\int_{-\pi}^{\pi} \sin(nt)\sin(mt) dt = \begin{cases} 0, & n \neq m \\ \pi, & n = m \end{cases}$ **Key insight**: $\cos(mt) \cdot \sin(nt)$ is an [odd function](https://en.wikipedia.org/wiki/Even_and_odd_functions), so: $\int_{-\pi}^{\pi} \cos(mt) \sin(nt) dt = 0$ **Result**: $a_m = \frac{1}{\pi} \int_{-\pi}^{\pi} f(t) \cos(mt) dt$ #### Method 3: Finding $b_n$ **Technique**: Multiply by $\sin(mt)$ and integrate **Result**: $b_m = \frac{1}{\pi} \int_{-\pi}^{\pi} f(t) \sin(mt) dt$ ### Part 4: Completing the Example For $f(t) = 2$: - $a_0 = 2$ - $a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} 2 \cos(nt) dt = 0$ (for all $n \geq 1$) - $b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} 2 \sin(nt) dt = 0$ (for all $n \geq 1$) **Therefore**: $f(t) = 2$ (as expected!) --- ## Mathematical Modeling Discussion ### Modeling Philosophy - Start with **wondering** ("I notice, I wonder" phase) - Initial wonderings often too large - need scoping - Balance between: - **Technical skills**: Mathematical techniques from curriculum - **Non-technical skills**: Problem formulation, communication - Mathematical modeling as our version of the scientific method - No laboratories, but we have: pencils, paper, computers, brains ### Student-Driven Projects - Models should reflect **your interests**, not the instructor's - Allows sustained engagement throughout semester - Past examples range from abstract (serial killer dynamics using [PDEs](https://en.wikipedia.org/wiki/Partial_differential_equation)) to creative (voice synthesis from skull models) - Connects to work from [[MATH310S26-Day1-Notes|Day 1]] on course objectives and [[MATH310S26-Day4-Notes|Day 4]] on modeling approaches --- ## Check Your Understanding ### Problem 1 Show that if $n, m \in \mathbb{N}$, then: $\int_{-\pi}^{\pi} \cos(nt)\cos(mt) dt = \begin{cases} 0, & n \neq m \\ \pi, & n = m \end{cases}$ $\int_{-\pi}^{\pi} \sin(nt)\sin(mt) dt = \begin{cases} 0, & n \neq m \\ \pi, & n = m \end{cases}$ ### Problem 2 Using the fact from Problem 1, show that if: $f(t) = a_0 + \sum_{n=1}^{\infty} [a_n \cos(nt) + b_n \sin(nt)]$ Then the coefficients are given by: - $a_0 = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) dt$ - $a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(t) \cos(nt) dt$ - $b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(t) \sin(nt) dt$ ### Problem 3 Using the formulas above, find $a_0, a_n, b_n$ for $f(t) = 2$. --- ## Key Concepts to Remember 1. **[Fourier series](https://en.wikipedia.org/wiki/Fourier_series)** decompose periodic functions into sums of sines and cosines 2. **[Orthogonality](https://en.wikipedia.org/wiki/Orthogonality)** of [trigonometric functions](https://en.wikipedia.org/wiki/Trigonometric_functions) is crucial for finding coefficients 3. **Integration** over a period isolates individual coefficients 4. Any "reasonable" periodic function can be represented as a Fourier series 5. **Periodic extension**: Even functions defined on finite intervals can be extended periodically --- ## Looking Ahead ### Next Class - Application of Fourier series to data analysis ### Monday's Focus - Low-stakes feedback assessment - Coding exercises with Fourier analysis (building on [[MATH310S26-Day3-WorkdayMaterials (Two-by-two matrices, normal equations, and ordinary least squares)|Day 3]] and [[MATH310S26-Day6-Work|Day 6]] coding work) - **Group formation for modeling projects** --- ## Related Mathematical Topics - **[Harmonic analysis](https://en.wikipedia.org/wiki/Harmonic_analysis)**: The broader field containing Fourier analysis - **[Spectral analysis](https://en.wikipedia.org/wiki/Spectral_analysis)**: Decomposing signals into frequency components - **[Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform)**: Extension to non-periodic functions - **[Principal Component Analysis](https://en.wikipedia.org/wiki/Principal_component_analysis)**: Related to [[MATH310S26-Day5-Notes|Day 5's eigenanalysis and dimensional reduction]] --- *Note: "We're cavalier" about switching limits in applied math - sometimes you just have to try things and see if they work!*