Fourier Analysis

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• Periodic phenomena and the idea of decomposition ~16 min
• Sines and cosines as building blocks ~16 min
• Time-domain vs. frequency-domain, a first picture ~15 min
• Fourier's original motivation: heat flow in a bar ~15 min
• A map of this course ~12 min
• Where Fourier analysis shows up ~14 min

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• Periodic functions and the fundamental period ~16 min
• Even and odd functions, and what symmetry buys you ~16 min
• Trigonometric polynomials ~15 min
• The inner product of two functions on an interval ~17 min
• Orthogonality of the sine/cosine family, proved directly ~20 min
• Norms and "energy" of a function ~17 min
• Building an orthonormal family from {sin, cos} ~16 min
• Why an orthogonal family is the right thing to expand a function in ~18 min

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• Deriving the coefficient formulas from orthogonality ~16 min
• The Fourier sine series (odd extension) ~14 min
• The Fourier cosine series (even extension) ~14 min
• The full Fourier series on a general interval ~13 min
• The complex form of the Fourier series ~15 min
• Converting between real and complex form via Euler's formula ~13 min
• Computing the series for a square wave ~15 min
• Computing the series for a sawtooth and a triangle wave ~15 min
• Half-range expansions ~14 min
• Symmetry shortcuts, revisited with real examples ~14 min

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• Sequences of functions, pointwise vs. uniform convergence ~16 min
• Why the distinction matters for a Fourier series specifically ~14 min
• Dirichlet's conditions ~16 min
• Convergence at points of continuity ~12 min
• Convergence at a jump discontinuity — the midpoint value ~14 min
• The Gibbs phenomenon ~15 min
• Measuring the overshoot quantitatively ~13 min
• Uniform convergence for smooth periodic functions ~15 min
• Cesàro (Fejér) averaging as a fix for non-uniform convergence ~12 min

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• Functions as vectors, revisited ~16 min
• Fourier coefficients as orthogonal projections ~18 min
• Bessel's inequality ~16 min
• Parseval's identity (energy conservation) ~18 min
• The L1, L2, and L∞ norms compared ~17 min
• Mean-square convergence ~16 min
• Completeness and Hilbert spaces ~17 min
• Why L2 is the right space for Fourier series ~15 min
• Density of trigonometric polynomials in L2 ~15 min
• What "best approximation" means for a truncated Fourier series ~16 min

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• Separation of variables, the core trick ~18 min
• The 1D heat equation: setup and boundary conditions ~16 min
• Solving the heat equation with a Fourier sine series ~20 min
• Long-term behavior and mode decay ~15 min
• The 1D wave equation and vibrating strings ~22 min
• Normal modes and standing waves ~16 min
• Laplace's equation on a rectangle ~20 min
• What a dedicated PDE course adds to this ~14 min

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• What breaks when a function isn't periodic ~12 min
• Taking the limit as the period grows without bound ~16 min
• The Fourier transform, defined ~14 min
• The inverse Fourier transform ~16 min
• Existence conditions — L¹ functions ~14 min
• Beyond the bound: transforms decay at high frequency ~14 min
• The frequency spectrum, continuous case ~13 min
• Reading a magnitude/phase plot ~13 min
• A first gallery: the transform of a simple pulse ~15 min

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• Linearity of the Fourier transform ~12 min
• Time-shifting and the phase-shift property ~12 min
• Frequency-shifting (modulation) ~12 min
• Scaling (time-compression/frequency-expansion duality) ~13 min
• Differentiation in time is multiplication in frequency ~14 min
• Duality ~13 min
• Plancherel's theorem: energy conservation for the transform ~14 min
• The Gaussian — its own transform ~15 min
• The rectangular pulse and the sinc function ~14 min
• The one-sided exponential ~13 min
• Motivating the delta function as a limit of narrowing pulses ~12 min
• The Dirac delta's defining (sifting) property ~12 min
• The Fourier transform of the delta, and of a constant ~14 min

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• The convolution integral, defined ~16 min
• The sliding-overlap geometric picture ~15 min
• Commutativity, associativity, and distributivity ~15 min
• Convolution as "blurring" or "smoothing," intuitively ~14 min
• The Convolution Theorem ~17 min
• Why the theorem makes filtering computationally cheap ~13 min
• Filtering as multiplication in the frequency domain ~15 min
• Low-pass, high-pass, and band-pass filters, defined by their frequency response ~16 min
• Applied problem set: signal smoothing and edge-preserving filtering ~16 min

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• Sampling a continuous signal ~15 min
• The Poisson summation bridge ~18 min
• Aliasing: two frequencies, one set of samples ~16 min
• The Nyquist–Shannon sampling theorem ~17 min
• Reconstruction from samples, informally ~15 min
• The discrete Fourier transform, defined ~16 min
• The DFT as multiplication by an explicit matrix ~15 min
• The inverse DFT ~16 min
• The DFT Parseval identity ~12 min
• Magnitude, phase, and conjugate symmetry for real signals ~15 min
• Circulant matrices and why the DFT basis diagonalizes them ~17 min
• Circular convolution and the discrete convolution theorem ~17 min

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• The cost of the naive DFT ~15 min
• Divide and conquer: splitting even and odd indices ~16 min
• The Cooley–Tukey recursion, derived ~18 min
• The butterfly operation ~17 min
• O(N²) vs. O(N log N), made concrete ~16 min
• Where the FFT matters in practice ~14 min

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• The 2D Fourier transform, defined ~18 min
• Separability — computing a 2D transform as 1D transforms ~17 min
• Radial symmetry and rotation ~17 min
• The 2D DFT ~18 min
• Spatial frequency, low vs. high, in an image ~17 min
• Applied problem set: frequency masking on a small synthetic test image ~16 min
• A brief look at diffraction as a 2D Fourier transform of an aperture ~16 min
• What Digital Image Processing adds to this ~15 min

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• The characteristic function of a random variable ~16 min
• Characteristic functions of standard distributions ~19 min
• Convolution of densities and products of characteristic functions ~18 min
• Sums of independent random variables, revisited ~18 min
• The Central Limit Theorem, through characteristic-function convergence ~22 min
• Recovering a density from its characteristic function ~17 min

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• Differentiation as an operator, and the Fourier basis as its eigenfunctions ~22 min
• Translation as an operator, and why the same basis diagonalizes it ~20 min
• Constant-coefficient differential equations become algebra in frequency space ~24 min
• The discrete case, revisited: circulant matrices and the DFT basis ~24 min
• Fourier series on the circle: the geometric picture behind periodicity ~20 min
• A unified recap: L¹/L², Parseval, and convergence as one landscape ~20 min
• What a dedicated digital signal processing course adds ~16 min
• What a dedicated physics course adds ~16 min
• What a dedicated PDE course adds: the operator picture, completed ~18 min