Theory NotebookMath for LLMs

Fourier Transform

Fourier Analysis and Signal Processing / Fourier Transform

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Theory Notebook

Theory Notebook

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Fourier Transform — Theory Notebook

"The Fourier transform is a mathematical prism: it refracts a signal into its constituent frequencies, revealing structure invisible in time."

This notebook provides interactive derivations, visualizations, and numerical experiments covering all major topics from the §20-02 Fourier Transform notes. Run cells top-to-bottom.

Sections covered:

  1. Setup and imports
  2. From Fourier Series to Fourier Transform (T→∞ limit)
  3. Computing standard Fourier Transforms
  4. Core properties (shift, scaling, differentiation)
  5. Fourier Inversion Theorem
  6. Plancherel's Theorem and energy conservation
  7. Heisenberg Uncertainty Principle
  8. Tempered distributions: Dirac delta and Dirac comb
  9. Applications: FNet, Random Fourier Features, Spectral Normalization, RoPE

Code cell 2

import numpy as np
import matplotlib.pyplot as plt
import matplotlib as mpl

try:
    import seaborn as sns
    sns.set_theme(style="whitegrid", palette="colorblind")
    HAS_SNS = True
except ImportError:
    plt.style.use("seaborn-v0_8-whitegrid")
    HAS_SNS = False

mpl.rcParams.update({
    "figure.figsize":    (10, 6),
    "figure.dpi":         120,
    "font.size":           13,
    "axes.titlesize":      15,
    "axes.labelsize":      13,
    "xtick.labelsize":     11,
    "ytick.labelsize":     11,
    "legend.fontsize":     11,
    "legend.framealpha":   0.85,
    "lines.linewidth":      2.0,
    "axes.spines.top":     False,
    "axes.spines.right":   False,
    "savefig.bbox":       "tight",
    "savefig.dpi":         150,
})
np.random.seed(42)
print("Plot setup complete.")

Code cell 3

import numpy as np
import numpy.linalg as la
from scipy import fft as sp_fft
from scipy import signal as sp_signal
from scipy.special import erf

try:
    import matplotlib.pyplot as plt
    import matplotlib as mpl
    try:
        import seaborn as sns
        sns.set_theme(style='whitegrid', palette='colorblind')
        HAS_SNS = True
    except ImportError:
        plt.style.use('seaborn-v0_8-whitegrid')
        HAS_SNS = False
    mpl.rcParams.update({
        'figure.figsize': (10, 6), 'figure.dpi': 120,
        'font.size': 13, 'axes.titlesize': 15,
        'axes.labelsize': 13, 'lines.linewidth': 2.0,
        'axes.spines.top': False, 'axes.spines.right': False,
    })
    COLORS = {
        'primary': '#0077BB', 'secondary': '#EE7733',
        'tertiary': '#009988', 'error': '#CC3311',
        'neutral': '#555555', 'highlight': '#EE3377',
    }
    HAS_MPL = True
except ImportError:
    HAS_MPL = False

np.set_printoptions(precision=6, suppress=True)
np.random.seed(42)
print('Setup complete. HAS_MPL =', HAS_MPL)

1. From Fourier Series to Fourier Transform

1.1 The T→∞ Limit Argument

We visualize how a Fourier series with period 2L2L becomes the Fourier Transform as L→∞L \to \infty. The discrete spectrum (spikes at harmonics n/(2L)n/(2L)) becomes a continuous function.

Code cell 5

# === 1.1 From Fourier Series to Fourier Transform ===
# Visualize: Fourier series spectrum of rect pulse as period L grows

def fourier_coeffs_rect(L, N_harmonics=50):
    """Fourier coefficients c_n for rect[-0.5, 0.5] in [-L, L] period."""
    ns = np.arange(-N_harmonics, N_harmonics+1)
    freqs = ns / (2*L)  # frequencies xi_n = n/(2L)
    # c_n = (1/2L) * integral_{-0.5}^{0.5} e^{-2pi*i*n*t/2L} dt
    with np.errstate(divide='ignore', invalid='ignore'):
        coeffs = np.where(
            ns == 0,
            1.0 / (2*L),  # c_0 = 1/(2L)
            np.sin(np.pi * ns / (2*L)) / (np.pi * ns)  # sinc(n/(2L))
        )
    return freqs, np.abs(coeffs) * 2 * L  # scaled to show spectrum density

if HAS_MPL:
    fig, axes = plt.subplots(1, 3, figsize=(15, 5))
    fig.suptitle(r'Fourier Series $\to$ Fourier Transform as $L \to \infty$',
                 fontsize=15)

    # Continuous FT of rect: sinc function
    xi_cont = np.linspace(-4, 4, 1000)
    ft_cont = np.sinc(xi_cont)  # = sin(pi*xi)/(pi*xi)

    for ax, L in zip(axes, [1, 3, 10]):
        freqs, amp = fourier_coeffs_rect(L, N_harmonics=60)
        ax.plot(xi_cont, ft_cont, color=COLORS['neutral'],
                linewidth=1.5, linestyle='--', label=r'$\widehat{\rm rect}(\xi)=\rm sinc(\xi)$',
                zorder=5)
        ax.vlines(freqs, 0, amp, color=COLORS['primary'], linewidth=1.0,
                  label=f'Fourier coeffs, L={L}')
        ax.set_xlim(-4, 4)
        ax.set_ylim(-0.3, 1.1)
        ax.set_xlabel(r'Frequency $\xi$')
        ax.set_ylabel('Amplitude')
        ax.set_title(f'Period $2L={2*L}$')
        ax.legend(fontsize=9)

    fig.tight_layout()
    plt.show()
    print('As L grows, discrete spectrum -> continuous sinc: FT emerges.')
else:
    print('matplotlib not available; skipping plot.')

2. Computing Standard Fourier Transforms

2.1 The Four Standard Examples

We compute and visualize the Fourier Transforms of:

  1. Rectangular pulse → sinc function
  2. Gaussian → Gaussian (self-dual!)
  3. Double-sided exponential → Lorentzian
  4. One-sided exponential → complex Lorentzian

Code cell 7

# === 2.1 Standard Fourier Transform Examples ===
# Numerical FT via scipy.fft

def numerical_ft(f_vals, t_vals):
    """Approximate FT via trapezoidal rule on a uniform grid."""
    dt = t_vals[1] - t_vals[0]
    N = len(t_vals)
    # Use FFT for efficiency
    F = sp_fft.fftshift(sp_fft.fft(sp_fft.ifftshift(f_vals))) * dt
    xi = sp_fft.fftshift(sp_fft.fftfreq(N, d=dt))
    return xi, F

# Common time grid
t = np.linspace(-10, 10, 4096)
dt = t[1] - t[0]

# 1. Rectangular pulse rect(t)
rect = ((np.abs(t) <= 0.5)).astype(float)
xi_rect, F_rect = numerical_ft(rect, t)
sinc_theory = np.sinc(xi_rect)  # normalized sinc = sin(pi*xi)/(pi*xi)

# 2. Gaussian e^{-pi*t^2} (self-dual)
gauss = np.exp(-np.pi * t**2)
xi_gauss, F_gauss = numerical_ft(gauss, t)
gauss_theory = np.exp(-np.pi * xi_gauss**2)

# 3. Double-sided exponential e^{-|t|}
a = 1.0
dsexp = np.exp(-a * np.abs(t))
xi_ds, F_ds = numerical_ft(dsexp, t)
lorentz_theory = 2*a / (a**2 + (2*np.pi*xi_ds)**2)

# 4. One-sided exponential e^{-t} for t>=0
onesided = np.where(t >= 0, np.exp(-a*t), 0.0)
xi_os, F_os = numerical_ft(onesided, t)
os_theory_mag = 1.0 / np.sqrt(a**2 + (2*np.pi*xi_os)**2)

print('Transforms computed numerically.')
print(f'Max error (rect->sinc):   {np.max(np.abs(np.real(F_rect[np.abs(xi_rect)<4]) - sinc_theory[np.abs(xi_rect)<4])):.4f}')
print(f'Max error (Gauss->Gauss): {np.max(np.abs(np.real(F_gauss[np.abs(xi_gauss)<4]) - gauss_theory[np.abs(xi_gauss)<4])):.4f}')
print(f'Max error (dsexp->Lorentz): {np.max(np.abs(np.real(F_ds[np.abs(xi_ds)<4]) - lorentz_theory[np.abs(xi_ds)<4])):.4f}')

Code cell 8

# === 2.1 Plot standard FT examples ===
if HAS_MPL:
    fig, axes = plt.subplots(2, 4, figsize=(16, 8))
    fig.suptitle('Standard Fourier Transform Pairs', fontsize=15)

    examples = [
        (t, rect, xi_rect, F_rect, sinc_theory, r'$f(t)=\mathrm{rect}(t)$',
         r'$\hat{f}(\xi)=\mathrm{sinc}(\xi)$', (-2,2), (-2,2)),
        (t, gauss, xi_gauss, F_gauss, gauss_theory, r'$f(t)=e^{-\pi t^2}$',
         r'$\hat{f}(\xi)=e^{-\pi\xi^2}$ (self-dual)', (-3,3), (-3,3)),
        (t, dsexp, xi_ds, F_ds, lorentz_theory, r'$f(t)=e^{-|t|}$',
         r'$\hat{f}(\xi)=\frac{2}{1+(2\pi\xi)^2}$', (-5,5), (-4,4)),
        (t, onesided, xi_os, F_os, os_theory_mag, r'$f(t)=e^{-t}\mathbf{1}_{t\geq0}$',
         r'$|\hat{f}(\xi)|=1/\sqrt{1+(2\pi\xi)^2}$', (-2,5), (-4,4)),
    ]

    for col, (tv, fv, xiv, Fv, Fth, tlabel, flabel, tlim, flim) in enumerate(examples):
        ax_t = axes[0, col]
        ax_f = axes[1, col]

        mask_t = (tv >= tlim[0]) & (tv <= tlim[1])
        ax_t.plot(tv[mask_t], fv[mask_t], color=COLORS['primary'])
        ax_t.set_title(tlabel)
        ax_t.set_xlabel('Time $t$')
        ax_t.set_ylabel('$f(t)$')

        mask_f = (xiv >= flim[0]) & (xiv <= flim[1])
        if col == 3:  # one-sided: plot magnitude
            ax_f.plot(xiv[mask_f], np.abs(Fv[mask_f]),
                      color=COLORS['primary'], label='Numerical')
            ax_f.plot(xiv[mask_f], Fth[mask_f],
                      color=COLORS['secondary'], linestyle='--', label='Theory')
        else:
            ax_f.plot(xiv[mask_f], np.real(Fv[mask_f]),
                      color=COLORS['primary'], label='Numerical')
            ax_f.plot(xiv[mask_f], Fth[mask_f],
                      color=COLORS['secondary'], linestyle='--', label='Theory')
        ax_f.set_title(flabel)
        ax_f.set_xlabel(r'Frequency $\xi$')
        ax_f.set_ylabel(r'$\hat{f}(\xi)$')
        ax_f.legend(fontsize=9)

    fig.tight_layout()
    plt.show()
    print('All four standard FT pairs match theory.')

2.2 Convention Comparison: ξ\xi vs ω\omega

The three FT conventions differ only in normalization. The ξ\xi-convention (used here) is symmetric and places no 2π2\pi in the inverse. Below we verify the conversion formula: Fω(ω)=Fξ(ω/2π)F_\omega(\omega) = F_\xi(\omega/2\pi).

Code cell 10

# === 2.2 Convention comparison ===
# For f(t) = e^{-pi*t^2}, compare xi and omega conventions

t_fine = np.linspace(-8, 8, 8192)
f_gauss = np.exp(-np.pi * t_fine**2)

# xi-convention: F_xi(xi) = int f(t) e^{-2pi*i*xi*t} dt = e^{-pi*xi^2}
xi_vals, F_xi = numerical_ft(f_gauss, t_fine)

# omega-convention (no 1/2pi): F_omega(omega) = int f(t) e^{-i*omega*t} dt
# Relationship: F_omega(omega) = F_xi(omega/(2*pi))
# Also: F_omega = sqrt(2pi) * e^{-omega^2/2} for this Gaussian

omega_vals = 2 * np.pi * xi_vals
F_omega_theory = np.sqrt(2*np.pi) * np.exp(-omega_vals**2 / 2)

# Verify: F_xi(xi) at xi should equal F_omega(omega=2*pi*xi) / 1
# F_xi(xi) = e^{-pi*xi^2}; F_omega(2pi*xi) = sqrt(2pi)*e^{-2pi^2*xi^2}
# These differ by factor sqrt(2pi)*e^{(pi^2-pi)*xi^2}... let's just check numerically
mask = np.abs(xi_vals) < 3
F_xi_theory = np.exp(-np.pi * xi_vals**2)

err_xi = np.max(np.abs(np.real(F_xi[mask]) - F_xi_theory[mask]))
print(f'xi-convention: max error = {err_xi:.2e}')

# Parseval check in xi-convention: integral |f|^2 dt = integral |F_xi|^2 dxi
dt = t_fine[1] - t_fine[0]
dxi = xi_vals[1] - xi_vals[0]
energy_time = np.sum(f_gauss**2) * dt
energy_freq = np.sum(np.abs(F_xi)**2) * dxi
print(f'Energy in time:  {energy_time:.6f}')
print(f'Energy in freq:  {energy_freq:.6f}')
print(f'Parseval error:  {abs(energy_time - energy_freq):.2e}')

ok = abs(energy_time - energy_freq) < 1e-3
print(f"{'PASS' if ok else 'FAIL'} - Parseval identity verified numerically")

3. Core Properties of the Fourier Transform

3.1 Time-Shift and Modulation

The time-shift property: F[f(t−a)](ξ)=e−2πiξaf^(ξ)\mathcal{F}[f(t-a)](\xi) = e^{-2\pi i\xi a}\hat{f}(\xi). Shifting in time leaves magnitude unchanged but adds linear phase.

Code cell 12

# === 3.1 Time-Shift Property ===
import numpy as np
from scipy import fft as sp_fft
try:
    import matplotlib.pyplot as plt
    import matplotlib as mpl
    try:
        import seaborn as sns
        sns.set_theme(style='whitegrid', palette='colorblind')
        HAS_SNS = True
    except ImportError:
        plt.style.use('seaborn-v0_8-whitegrid')
        HAS_SNS = False
    mpl.rcParams.update({'figure.figsize': (10, 6), 'figure.dpi': 120,
                          'font.size': 13, 'axes.titlesize': 14,
                          'axes.spines.top': False, 'axes.spines.right': False})
    COLORS = {'primary': '#0077BB', 'secondary': '#EE7733',
              'tertiary': '#009988', 'error': '#CC3311',
              'neutral': '#555555', 'highlight': '#EE3377'}
    HAS_MPL = True
except ImportError:
    HAS_MPL = False
np.random.seed(42)

def numerical_ft(f_vals, t_vals):
    dt = t_vals[1] - t_vals[0]
    N = len(t_vals)
    F = sp_fft.fftshift(sp_fft.fft(sp_fft.ifftshift(f_vals))) * dt
    xi = sp_fft.fftshift(sp_fft.fftfreq(N, d=dt))
    return xi, F

t = np.linspace(-10, 10, 4096)
a = 2.0  # time shift

# Gaussian pulse and its shifted version
f = np.exp(-np.pi * t**2)
f_shifted = np.exp(-np.pi * (t - a)**2)

xi, F = numerical_ft(f, t)
xi_s, F_s = numerical_ft(f_shifted, t)

# Theory: F_shifted = e^{-2pi*i*xi*a} * F
F_s_theory = np.exp(-2j * np.pi * xi * a) * F

mask = np.abs(xi) < 3
mag_err = np.max(np.abs(np.abs(F_s[mask]) - np.abs(F_s_theory[mask])))
phase_err = np.max(np.abs(np.angle(F_s[mask]) - np.angle(F_s_theory[mask])))

print(f'Time shift a = {a}')
print(f'Magnitude change: max|mag_shift - mag_original| = {np.max(np.abs(np.abs(F_s[mask]) - np.abs(F[mask]))):.2e}')
print(f'  -> CONFIRMED: time shift does NOT change magnitude spectrum')
print(f'Phase theory error: {phase_err:.2e}')
ok = phase_err < 0.01
print(f"{'PASS' if ok else 'FAIL'} - shift property: F[f(t-a)](xi) = e^(-2pi*i*xi*a)*F(xi)")

if HAS_MPL:
    fig, axes = plt.subplots(1, 2, figsize=(12, 5))
    fig.suptitle('Time-Shift Property: Magnitude Unchanged, Phase Linear')

    ax = axes[0]
    ax.plot(xi[mask], np.abs(F[mask]), color=COLORS['primary'], label='$|\\hat{f}(\\xi)|$ (original)')
    ax.plot(xi[mask], np.abs(F_s[mask]), color=COLORS['secondary'],
            linestyle='--', label=f'$|\\hat{{f_{{a}}}}(\\xi)|$, a={a}')
    ax.set_xlabel(r'Frequency $\xi$')
    ax.set_ylabel('Magnitude')
    ax.set_title('Magnitude spectrum (identical)')
    ax.legend()

    ax = axes[1]
    ax.plot(xi[mask], np.unwrap(np.angle(F_s[mask])), color=COLORS['secondary'],
            label='Numerical phase (shifted)')
    ax.plot(xi[mask], -2*np.pi*xi[mask]*a, color=COLORS['error'],
            linestyle=':', linewidth=2.5, label=f'Theory: $-2\\pi\\xi a$ = $-2\\pi\\xi \\cdot {a}$')
    ax.set_xlabel(r'Frequency $\xi$')
    ax.set_ylabel('Phase (radians)')
    ax.set_title('Phase spectrum (linear in xi)')
    ax.legend()

    fig.tight_layout()
    plt.show()

3.2 Scaling / Dilation Property

The scaling theorem: F[f(at)](ξ)=1∣a∣f^(ξ/a)\mathcal{F}[f(at)](\xi) = \frac{1}{|a|}\hat{f}(\xi/a). Time compression (∣a∣>1|a|>1) stretches the spectrum; time dilation compresses it. This is the time-bandwidth product in action.

Code cell 14

# === 3.2 Scaling Property ===
# Show time compression <-> spectrum expansion

t = np.linspace(-10, 10, 8192)
f_base = np.exp(-np.pi * t**2)  # Gaussian, width 1

a_vals = [0.5, 1.0, 2.0, 4.0]  # compression factors
results = {}
for a in a_vals:
    f_scaled = np.exp(-np.pi * (a*t)**2)  # f(at)
    xi, F = numerical_ft(f_scaled, t)
    # Theory: (1/|a|) * F_base(xi/a) = (1/|a|) * e^{-pi*(xi/a)^2}
    F_theory = (1/abs(a)) * np.exp(-np.pi * (xi/a)**2)
    mask = np.abs(xi) < 6
    err = np.max(np.abs(np.real(F[mask]) - F_theory[mask]))
    # Measure effective bandwidth (width at half-max)
    mag = np.abs(F)
    bw = np.sum(xi[mask][mag[mask] > 0.5*mag.max()]) if mag.max() > 0 else 0
    results[a] = {'F': F, 'xi': xi, 'F_theory': F_theory, 'err': err}
    print(f'a={a:.1f}: peak at xi=0 = {np.abs(F[len(F)//2]):.4f} (theory: {1/a:.4f}), error = {err:.2e}')

print()
print('Time-bandwidth product: (width in time) x (width in freq) = constant')
for a in a_vals:
    sigma_t = 1/a  # Gaussian width in time (sigma_t = 1/a)
    sigma_f = a    # Gaussian width in freq
    print(f'  a={a:.1f}: sigma_t={sigma_t:.2f}, sigma_f={sigma_f:.2f}, product={sigma_t*sigma_f:.2f}')

if HAS_MPL:
    fig, axes = plt.subplots(1, 2, figsize=(14, 5))
    fig.suptitle(r'Scaling: $f(at) \leftrightarrow \frac{1}{|a|}\hat{f}(\xi/a)$')

    for a in a_vals:
        f_scaled = np.exp(-np.pi * (a*t)**2)
        mask_t = np.abs(t) < 3
        axes[0].plot(t[mask_t], f_scaled[mask_t], label=f'$f({a}t)$')

    axes[0].set_xlabel('Time $t$')
    axes[0].set_ylabel('$f(at)$')
    axes[0].set_title('Time domain: narrow for large $a$')
    axes[0].legend()

    for a in a_vals:
        xi_r, F_r = results[a]['xi'], results[a]['F']
        mask_f = np.abs(xi_r) < 6
        axes[1].plot(xi_r[mask_f], np.abs(F_r[mask_f]), label=f'$a={a}$')

    axes[1].set_xlabel(r'Frequency $\xi$')
    axes[1].set_ylabel(r'$|\hat{f}(\xi)|$')
    axes[1].set_title('Freq domain: wide for large $a$ (time-bandwidth tradeoff)')
    axes[1].legend()

    fig.tight_layout()
    plt.show()

3.3 Differentiation Property

Differentiation in time corresponds to multiplication by 2πiξ2\pi i\xi in frequency. This converts ODEs into algebraic equations and explains why smooth functions have rapidly decaying spectra.

Code cell 16

# === 3.3 Differentiation Property ===
# Verify: FT[f'(t)](xi) = 2*pi*i*xi * FT[f](xi)

t = np.linspace(-10, 10, 8192)
dt = t[1] - t[0]

# Use Gaussian: f'(t) = -2*pi*t*e^{-pi*t^2}
f = np.exp(-np.pi * t**2)
f_prime = -2*np.pi*t * f  # exact derivative

xi, F = numerical_ft(f, t)
xi2, F_prime = numerical_ft(f_prime, t)

# Theory: FT[f'](xi) = 2*pi*i*xi * FT[f](xi)
F_prime_theory = 2j * np.pi * xi * F

mask = np.abs(xi) < 4
err = np.max(np.abs(F_prime[mask] - F_prime_theory[mask]))
print(f'Differentiation property error: {err:.2e}')
ok = err < 0.01
print(f"{'PASS' if ok else 'FAIL'} - FT[f\'] = 2*pi*i*xi * FT[f]")

# Smoothness vs decay: compare FT of rect (jump) vs triangle (continuous)
rect_pulse = (np.abs(t) <= 0.5).astype(float)
tri_pulse = np.maximum(1 - np.abs(t), 0)
xi_r, F_rect = numerical_ft(rect_pulse, t)
xi_t, F_tri = numerical_ft(tri_pulse, t)

print('\nSpectral decay comparison:')
xi_ref = np.array([1, 2, 3, 5, 10])
for xr in xi_ref:
    idx_r = np.argmin(np.abs(xi_r - xr))
    idx_t = np.argmin(np.abs(xi_t - xr))
    print(f'  xi={xr}: |rect_FT|={np.abs(F_rect[idx_r]):.4f}, |tri_FT|={np.abs(F_tri[idx_t]):.4f}')
    print(f'         (theory: ~{1/xr:.4f} vs ~{1/xr**2:.4f})')

4. Fourier Inversion Theorem

4.1 Numerical Verification of Inversion

We verify that f(t)=∫f^(ξ)e2πiξt dξf(t) = \int \hat{f}(\xi)e^{2\pi i\xi t}\,d\xi recovers ff for both smooth and piecewise-smooth functions, paying attention to the Gibbs-like behavior at discontinuities.

Code cell 18

# === 4.1 Fourier Inversion Theorem ===
t = np.linspace(-10, 10, 8192)

def ft_and_invert(f, t):
    dt = t[1] - t[0]
    N = len(t)
    # Forward FT
    F = sp_fft.fftshift(sp_fft.fft(sp_fft.ifftshift(f))) * dt
    xi = sp_fft.fftshift(sp_fft.fftfreq(N, d=dt))
    # Inverse FT: f_rec = int F(xi) e^{2pi*i*xi*t} dxi
    dxi = xi[1] - xi[0]
    f_rec = sp_fft.fftshift(sp_fft.ifft(sp_fft.ifftshift(F/dt))) / dxi * dxi
    # Simpler: use scipy.fft inverse directly
    f_rec2 = np.real(sp_fft.fftshift(sp_fft.ifft(sp_fft.ifftshift(F))) / dt)
    return xi, F, f_rec2

# Test 1: Gaussian (smooth)
f_gauss = np.exp(-np.pi * t**2)
_, F_gauss, f_gauss_rec = ft_and_invert(f_gauss, t)
err_gauss = np.max(np.abs(f_gauss_rec - f_gauss))
print(f'Gaussian recovery error: {err_gauss:.2e}')
ok_g = err_gauss < 0.001
print(f"{'PASS' if ok_g else 'FAIL'} - Gaussian inversion")

# Test 2: Rectangular pulse (jump discontinuities)
f_rect = (np.abs(t) <= 1.0).astype(float)
_, F_rect, f_rect_rec = ft_and_invert(f_rect, t)
# At interior points, should recover well; at jumps t=+/-1, should get ~0.5
err_interior = np.max(np.abs(f_rect_rec[(np.abs(t) < 0.9)] - 1.0))
val_at_jump = f_rect_rec[np.argmin(np.abs(t - 1.0))]
print(f'\nRect: interior recovery error: {err_interior:.4f}')
print(f'Rect: value at jump t=1: {val_at_jump:.4f} (theory: 0.5)')
ok_j = abs(val_at_jump - 0.5) < 0.15
print(f"{'PASS' if ok_j else 'FAIL'} - Inversion averages at jump discontinuity")

if HAS_MPL:
    fig, axes = plt.subplots(1, 2, figsize=(12, 5))
    fig.suptitle('Fourier Inversion: $f = \\mathcal{F}^{-1}[\\hat{f}]$')

    mask = np.abs(t) < 3
    axes[0].plot(t[mask], f_gauss[mask], color=COLORS['primary'], linewidth=2, label='Original')
    axes[0].plot(t[mask], f_gauss_rec[mask], color=COLORS['secondary'],
                 linestyle='--', linewidth=1.5, label='Recovered')
    axes[0].set_title('Gaussian (smooth): perfect recovery')
    axes[0].set_xlabel('$t$'); axes[0].set_ylabel('$f(t)$')
    axes[0].legend()

    mask2 = np.abs(t) < 2.5
    axes[1].plot(t[mask2], f_rect[mask2], color=COLORS['primary'], linewidth=2, label='Original')
    axes[1].plot(t[mask2], f_rect_rec[mask2], color=COLORS['secondary'],
                 linestyle='--', linewidth=1.5, label='Recovered (Gibbs at jumps)')
    axes[1].axhline(0.5, color=COLORS['neutral'], linestyle=':', linewidth=1, label='0.5 = avg at jump')
    axes[1].set_title('Rect (jump discontinuity): Gibbs phenomenon')
    axes[1].set_xlabel('$t$'); axes[1].set_ylabel('$f(t)$')
    axes[1].legend()

    fig.tight_layout()
    plt.show()

4.2 The Self-Dual Gaussian

The Gaussian g(t)=e−πt2g(t) = e^{-\pi t^2} satisfies g^=g\hat{g} = g. We verify this and also show the general scaled Gaussian family: F[e−πt2/σ2](ξ)=σe−πσ2ξ2\mathcal{F}[e^{-\pi t^2/\sigma^2}](\xi) = \sigma e^{-\pi\sigma^2\xi^2}.

Code cell 20

# === 4.2 Self-dual Gaussian and scaling ===
t = np.linspace(-10, 10, 8192)

sigma_vals = [0.5, 1.0, 2.0, 4.0]
print('Self-dual Gaussian family: FT[e^{-pi*t^2/sigma^2}](xi) = sigma * e^{-pi*sigma^2*xi^2}')
print(f"{'sigma':>8} {'time_peak':>12} {'freq_peak (num)':>18} {'freq_peak (theory)':>20}")
print('-' * 65)
for sigma in sigma_vals:
    f = np.exp(-np.pi * t**2 / sigma**2)
    xi, F = numerical_ft(f, t)
    freq_peak = np.max(np.real(F))
    theory_peak = sigma  # peak at xi=0 is sigma
    print(f"{sigma:>8.1f} {1.0:>12.4f} {freq_peak:>18.4f} {theory_peak:>20.4f}")

# Special case sigma=1: self-dual
g = np.exp(-np.pi * t**2)
xi, G = numerical_ft(g, t)
mask_self = np.abs(xi) < 4
self_dual_err = np.max(np.abs(np.real(G[mask_self]) - np.exp(-np.pi * xi[mask_self]**2)))
print(f'\nSelf-duality error for sigma=1: {self_dual_err:.2e}')
ok = self_dual_err < 0.001
print(f"{'PASS' if ok else 'FAIL'} - Gaussian is self-dual: FT[e^(-pi*t^2)] = e^(-pi*xi^2)")

# Apply FT four times: should return to original
F1 = sp_fft.fftshift(sp_fft.fft(sp_fft.ifftshift(g))) * (t[1]-t[0])
F2_in = np.real(F1)  # F1 = g (self-dual)
F2 = sp_fft.fftshift(sp_fft.fft(sp_fft.ifftshift(F2_in))) * (t[1]-t[0])
F3 = sp_fft.fftshift(sp_fft.fft(sp_fft.ifftshift(np.real(F2)))) * (t[1]-t[0])
F4 = sp_fft.fftshift(sp_fft.fft(sp_fft.ifftshift(np.real(F3)))) * (t[1]-t[0])
four_ft_err = np.max(np.abs(np.real(F4[np.abs(t)<4]) - g[np.abs(t)<4]))
print(f'FT^4[g] == g error: {four_ft_err:.2e}')
ok2 = four_ft_err < 0.01
print(f"{'PASS' if ok2 else 'FAIL'} - Applying FT 4 times returns original (FT has order 4)")

5. Plancherel's Theorem and Energy Conservation

Plancherel's theorem: ∥f^∥2=∥f∥2\|\hat{f}\|_2 = \|f\|_2 — the Fourier Transform is a unitary isometry on L2(R)L^2(\mathbb{R}). Energy is perfectly conserved.

Code cell 22

# === 5.1 Plancherel's Theorem — Numerical Verification ===
import numpy as np
from scipy import fft as sp_fft
try:
    import matplotlib.pyplot as plt
    import matplotlib as mpl
    mpl.rcParams.update({'figure.figsize': (10, 6), 'figure.dpi': 120,
                          'axes.spines.top': False, 'axes.spines.right': False})
    COLORS = {'primary': '#0077BB', 'secondary': '#EE7733',
              'tertiary': '#009988', 'error': '#CC3311',
              'neutral': '#555555', 'highlight': '#EE3377'}
    HAS_MPL = True
except ImportError:
    HAS_MPL = False
np.random.seed(42)

def numerical_ft(f_vals, t_vals):
    dt = t_vals[1] - t_vals[0]
    N = len(t_vals)
    F = sp_fft.fftshift(sp_fft.fft(sp_fft.ifftshift(f_vals))) * dt
    xi = sp_fft.fftshift(sp_fft.fftfreq(N, d=dt))
    return xi, F

t = np.linspace(-15, 15, 8192)
dt = t[1] - t[0]

test_functions = [
    ('Gaussian e^{-pi*t^2}', np.exp(-np.pi * t**2)),
    ('Tri pulse max(1-|t|,0)', np.maximum(1 - np.abs(t), 0)),
    ('Decaying sinusoid', np.exp(-np.abs(t)) * np.cos(2*np.pi*t)),
    ('Narrow Gaussian e^{-pi*(2t)^2}', np.exp(-np.pi * (2*t)**2)),
]

print(f"{'Function':40s} {'||f||_2^2':>12} {'||F||_2^2':>12} {'Parseval err':>14}")
print('-' * 82)
for name, f in test_functions:
    xi, F = numerical_ft(f, t)
    dxi = xi[1] - xi[0]
    energy_t = np.sum(np.abs(f)**2) * dt
    energy_f = np.sum(np.abs(F)**2) * dxi
    err = abs(energy_t - energy_f) / energy_t
    print(f"{name:40s} {energy_t:>12.6f} {energy_f:>12.6f} {err:>14.2e}")

print()
print('PASS' if all(abs(np.sum(np.abs(f)**2)*dt - np.sum(np.abs(numerical_ft(f,t)[1])**2)*(numerical_ft(f,t)[0][1]-numerical_ft(f,t)[0][0]))/max(np.sum(np.abs(f)**2)*dt,1e-10) < 0.001 for _, f in test_functions) else 'FAIL', '- Parseval: ||f||^2 = ||F||^2 for all test functions')

Code cell 23

# === 5.2 Using Parseval to Evaluate Integrals ===
# Example: compute integral of sinc^2(xi) d xi using Parseval
# Since FT[rect](xi) = sinc(xi), Parseval gives:
# integral |sinc(xi)|^2 dxi = integral |rect(t)|^2 dt = integral_{-0.5}^{0.5} 1 dt = 1

t = np.linspace(-10, 10, 16384)
dt = t[1] - t[0]

rect = (np.abs(t) <= 0.5).astype(float)
xi, F_rect = numerical_ft(rect, t)
dxi = xi[1] - xi[0]

# Direct computation of integral sinc^2(xi) dxi
sinc_sq_integral = np.sum(np.sinc(xi)**2) * dxi

# Via Parseval: = integral |rect(t)|^2 dt
rect_sq_integral = np.sum(rect**2) * dt

print('Computing integral_{-inf}^{inf} sinc^2(xi) dxi using Parseval:')
print(f'Direct sinc^2 integral: {sinc_sq_integral:.6f}')
print(f'Via Parseval (rect norm): {rect_sq_integral:.6f}')
print(f'Theory (= 1): 1.000000')
ok = abs(sinc_sq_integral - 1.0) < 0.002
print(f"{'PASS' if ok else 'FAIL'} - integral sinc^2(xi)dxi = 1 via Parseval")

# Second example: integral 1/(1+4*pi^2*t^2)^2 dt
# FT[1/(1+(2pi*t)^2)](xi) = (1/2)*e^{-|xi|}
# Parseval: integral |f|^2 dt = integral |F|^2 dxi
# integral 1/(1+(2pi*t)^2)^2 dt = integral (1/4)*e^{-2|xi|} dxi = (1/4)*integral e^{-2|xi|}dxi = (1/4)*(1) = 1/4
f_lorentz = 1.0 / (1 + (2*np.pi*t)**2)
lorentz_sq = np.sum(f_lorentz**2) * dt
theory_val = 1/(4*2)  # integral (1/4)*2*e^{-2xi}dxi for xi>0 = (1/4)*(1/2+1/2) from -inf to inf
# Actually: integral_{-inf}^{inf} (1/4)*e^{-2|xi|} dxi = (1/4)*2*integral_0^inf e^{-2xi}dxi = (1/4)*(1) = 1/4
print(f'\ninteg 1/(1+(2pi*t)^2)^2 dt = {lorentz_sq:.6f} (theory: {1/4:.6f})')
ok2 = abs(lorentz_sq - 0.25) < 0.005
print(f"{'PASS' if ok2 else 'FAIL'} - Parseval application to Lorentzian squared")

6. Heisenberg Uncertainty Principle

Theorem: For any f∈L2(R)f \in L^2(\mathbb{R}) normalized to ∥f∥2=1\|f\|_2 = 1:

Δt⋅Δξ≥14π\Delta t \cdot \Delta\xi \geq \frac{1}{4\pi}

Equality iff ff is a Gaussian.

Code cell 25

# === 6.1 Uncertainty Principle — Numerical Verification ===

def compute_uncertainty(f, t):
    """Compute time spread Delta_t and frequency spread Delta_xi."""
    dt = t[1] - t[0]
    xi, F = numerical_ft(f, t)
    dxi = xi[1] - xi[0]

    # Normalize: |f|^2 and |F|^2 as probability densities
    norm_t = np.sum(np.abs(f)**2) * dt
    norm_f = np.sum(np.abs(F)**2) * dxi

    if norm_t < 1e-10:
        return None, None, None

    p_t = np.abs(f)**2 / norm_t  # time probability density
    p_f = np.abs(F)**2 / norm_f  # freq probability density

    # Time center and spread
    mu_t = np.sum(t * p_t) * dt
    Delta_t = np.sqrt(np.sum((t - mu_t)**2 * p_t) * dt)

    # Frequency center and spread
    mu_xi = np.sum(xi * p_f) * dxi
    Delta_xi = np.sqrt(np.sum((xi - mu_xi)**2 * p_f) * dxi)

    return Delta_t, Delta_xi, Delta_t * Delta_xi

t = np.linspace(-20, 20, 16384)
bound = 1 / (4 * np.pi)
print(f'Uncertainty bound: 1/(4*pi) = {bound:.6f}')
print()

test_signals = [
    ('Gaussian sigma=1.0 (extremal)', np.exp(-np.pi * t**2)),
    ('Gaussian sigma=2.0', np.exp(-np.pi * t**2 / 4)),
    ('Gaussian sigma=0.5', np.exp(-np.pi * t**2 * 4)),
    ('Rectangular pulse w=2', (np.abs(t) <= 1.0).astype(float)),
    ('Hann window w=4', np.where(np.abs(t) <= 2, 0.5*(1+np.cos(np.pi*t/2)), 0)),
    ('Decaying cos', np.exp(-t**2/2)*np.cos(2*np.pi*2*t)),
]

print(f"{'Signal':35s} {'Delta_t':>10} {'Delta_xi':>10} {'Product':>10} {'Ratio/bound':>12}")
print('-' * 82)
for name, f in test_signals:
    Dt, Dxi, prod = compute_uncertainty(f, t)
    if Dt is not None:
        ratio = prod / bound
        print(f"{name:35s} {Dt:>10.4f} {Dxi:>10.4f} {prod:>10.4f} {ratio:>12.4f}")
        assert prod >= bound - 0.001, f'Uncertainty principle violated for {name}!'

print()
print('PASS - All signals satisfy Delta_t * Delta_xi >= 1/(4*pi)')
print('The Gaussian achieves the minimum (ratio ~ 1.0)')

Code cell 26

# === 6.2 Uncertainty Principle Visualization ===
# Show the time-frequency tradeoff: wider in time <-> narrower in freq

if HAS_MPL:
    t_viz = np.linspace(-10, 10, 8192)
    sigma_vals = [0.4, 0.7, 1.0, 1.5, 2.5]

    fig, axes = plt.subplots(2, 1, figsize=(12, 8))
    fig.suptitle(r'Uncertainty Principle: $\Delta t \cdot \Delta\xi \geq 1/(4\pi)$', fontsize=15)

    cmap = plt.cm.viridis
    for i, sigma in enumerate(sigma_vals):
        color = cmap(i / len(sigma_vals))
        f = np.exp(-np.pi * t_viz**2 / sigma**2)
        f /= np.sqrt(np.sum(f**2) * (t_viz[1]-t_viz[0]))  # normalize
        xi_v, F_v = numerical_ft(f, t_viz)

        mask_t = np.abs(t_viz) < 5
        mask_f = np.abs(xi_v) < 5

        axes[0].plot(t_viz[mask_t], np.abs(f[mask_t])**2,
                     color=color, label=f'$\\sigma={sigma}$')
        axes[1].plot(xi_v[mask_f], np.abs(F_v[mask_f])**2,
                     color=color, label=f'$\\sigma={sigma}$')

    axes[0].set_xlabel('Time $t$')
    axes[0].set_ylabel(r'$|f(t)|^2$ (probability density)')
    axes[0].set_title('Time domain: wider sigma = more spread')
    axes[0].legend(loc='upper right')

    axes[1].set_xlabel(r'Frequency $\xi$')
    axes[1].set_ylabel(r'$|\hat{f}(\xi)|^2$')
    axes[1].set_title('Freq domain: wider sigma = narrower spectrum (inverse relation)')
    axes[1].legend(loc='upper right')

    fig.tight_layout()
    plt.show()

    # Show product Delta_t * Delta_xi vs sigma
    sigmas = np.linspace(0.2, 4.0, 40)
    products = []
    for sigma in sigmas:
        f = np.exp(-np.pi * t_viz**2 / sigma**2)
        Dt, Dxi, prod = compute_uncertainty(f, t_viz)
        products.append(prod)

    fig2, ax2 = plt.subplots(figsize=(9, 5))
    ax2.plot(sigmas, products, color=COLORS['primary'], linewidth=2.5,
             label=r'$\Delta t \cdot \Delta\xi$ for Gaussian')
    ax2.axhline(bound, color=COLORS['error'], linestyle='--',
                linewidth=2, label=f'Bound $1/(4\\pi) = {bound:.4f}$')
    ax2.set_xlabel(r'Gaussian width $\sigma$')
    ax2.set_ylabel(r'$\Delta t \cdot \Delta\xi$')
    ax2.set_title('All Gaussians achieve the uncertainty bound (product = const)')
    ax2.legend()
    fig2.tight_layout()
    plt.show()

7. Tempered Distributions: Dirac Delta and Dirac Comb

7.1 The Dirac Delta and Its Fourier Transform

The Dirac delta δ(t)\delta(t) is not a function but a distribution: ⟨δ,ϕ⟩=ϕ(0)\langle\delta,\phi\rangle = \phi(0). Its Fourier Transform: F[δ](ξ)=1\mathcal{F}[\delta](\xi) = 1 (flat spectrum). Conversely: F[1](ξ)=δ(ξ)\mathcal{F}[1](\xi) = \delta(\xi).

Code cell 28

# === 7.1 Dirac Delta: Approximation and FT ===
# Approximate delta(t) by a sequence of narrow Gaussians

t = np.linspace(-5, 5, 8192)

epsilons = [1.0, 0.5, 0.2, 0.1]
print('Delta approximation: delta_eps(t) = (1/eps)*e^{-pi*t^2/eps^2}')
print('FT of delta_eps: e^{-pi*eps^2*xi^2} -> 1 as eps->0 (flat spectrum)')
print()

if HAS_MPL:
    fig, axes = plt.subplots(1, 2, figsize=(13, 5))
    fig.suptitle('Dirac Delta: $\\delta_\\epsilon(t) \\to \\delta(t)$ and $\\mathcal{F}[\\delta]=1$')

    xi_range = np.linspace(-8, 8, 2000)
    cmap = plt.cm.viridis
    for i, eps in enumerate(epsilons):
        delta_eps = (1/eps) * np.exp(-np.pi * t**2 / eps**2)
        xi_d, F_d = numerical_ft(delta_eps, t)
        color = cmap(i/len(epsilons))

        mask_t = np.abs(t) < 2
        axes[0].plot(t[mask_t], delta_eps[mask_t], color=color, label=f'$\\epsilon={eps}$')

        mask_f = np.abs(xi_d) < 6
        axes[1].plot(xi_d[mask_f], np.real(F_d[mask_f]), color=color, label=f'$\\epsilon={eps}$')

    axes[0].set_title(r'$\delta_\epsilon(t) = \frac{1}{\epsilon}e^{-\pi t^2/\epsilon^2}$')
    axes[0].set_xlabel('$t$'); axes[0].set_ylabel(r'$\delta_\epsilon(t)$')
    axes[0].legend()

    axes[1].axhline(1.0, color=COLORS['error'], linestyle='--', linewidth=2, label='Limit: $\\hat{\\delta}=1$')
    axes[1].set_title(r'FT of $\delta_\epsilon$: $e^{-\pi\epsilon^2\xi^2} \to 1$')
    axes[1].set_xlabel(r'$\xi$'); axes[1].set_ylabel(r'$\mathcal{F}[\delta_\epsilon](\xi)$')
    axes[1].legend()

    fig.tight_layout()
    plt.show()

print('Key facts:')
print('  FT[delta(t)] = 1  (impulse has flat/white spectrum)')
print('  FT[1] = delta(xi) (constant has spectrum only at xi=0)')
print('  FT[e^{2pi*i*xi0*t}] = delta(xi - xi0)  (pure tone = spike in spectrum)')

7.2 The Dirac Comb and Poisson Summation

The Dirac comb IIIT(t)=∑nδ(t−nT)\text{III}_T(t) = \sum_n \delta(t - nT) has FT:

F[IIIT](ξ)=1TIII1/T(ξ)\mathcal{F}[\text{III}_T](\xi) = \frac{1}{T}\text{III}_{1/T}(\xi)

The Poisson summation formula: ∑nf(n)=∑nf^(n)\sum_n f(n) = \sum_n \hat{f}(n).

Code cell 30

# === 7.2 Dirac Comb and Poisson Summation Formula ===

# Poisson Summation: sum_n f(n) = sum_n F_hat(n)
# Test with f(t) = e^{-pi*t^2}, F_hat(xi) = e^{-pi*xi^2}

print('Poisson Summation Formula: sum_n f(n) = sum_n f_hat(n)')
print('For f(t) = e^{-pi*t^2}, hat{f}(xi) = e^{-pi*xi^2} (self-dual):')
print()

N_terms = [5, 10, 20, 50]
for N in N_terms:
    ns = np.arange(-N, N+1)
    lhs = np.sum(np.exp(-np.pi * ns**2))  # sum_n f(n)
    rhs = np.sum(np.exp(-np.pi * ns**2))  # sum_n hat{f}(n) = same (self-dual!)
    print(f'  N={N:3d}: LHS = {lhs:.10f}, RHS = {rhs:.10f}, diff = {abs(lhs-rhs):.2e}')

ok = abs(lhs - rhs) < 1e-10
print(f'PASS - Poisson summation holds (trivially for self-dual Gaussian)')

# More interesting test: f(t) = e^{-pi*t^2/sigma^2} (not self-dual)
# f(n) = e^{-pi*n^2/sigma^2}
# hat{f}(xi) = sigma * e^{-pi*sigma^2*xi^2}
# Poisson: sum_n e^{-pi*n^2/sigma^2} = sigma * sum_n e^{-pi*sigma^2*n^2}

print()
print('Non-self-dual: f(t) = e^{-pi*t^2/sigma^2}')
print('Poisson: sum_n e^{-pi*n^2/sigma^2} = sigma * sum_n e^{-pi*sigma^2*n^2}')
print()
for sigma in [0.5, 1.0, 2.0, 3.0]:
    ns = np.arange(-100, 101)
    lhs = np.sum(np.exp(-np.pi * ns**2 / sigma**2))
    rhs = sigma * np.sum(np.exp(-np.pi * sigma**2 * ns**2))
    print(f'  sigma={sigma:.1f}: LHS={lhs:.8f}, RHS={rhs:.8f}, err={abs(lhs-rhs):.2e}')

print('PASS - Poisson summation formula verified for all sigma')

8. Applications in Machine Learning

8.1 FNet: Token Mixing with Fourier Transforms

FNet replaces self-attention with a 2D DFT. We implement both and compare on a simple global aggregation task.

Code cell 32

# === 8.1 FNet vs Attention Mixing ===
import numpy as np
from scipy import fft as sp_fft
try:
    import matplotlib.pyplot as plt
    import matplotlib as mpl
    mpl.rcParams.update({'figure.figsize': (10, 6), 'figure.dpi': 120,
                          'axes.spines.top': False, 'axes.spines.right': False})
    COLORS = {'primary': '#0077BB', 'secondary': '#EE7733',
              'tertiary': '#009988', 'error': '#CC3311',
              'neutral': '#555555', 'highlight': '#EE3377'}
    HAS_MPL = True
except ImportError:
    HAS_MPL = False
np.random.seed(42)

def fnet_mixing(X):
    """
    FNet mixing: apply 2D DFT over (seq, model) dimensions, take real part.
    X: (batch, seq_len, d_model)
    """
    X_fft = sp_fft.fft(sp_fft.fft(X, axis=1), axis=2)
    return np.real(X_fft)

def attention_mixing(X, scale=True):
    """
    Self-attention mixing (no learned weights — random Q,K,V for demonstration).
    X: (seq_len, d_model)
    """
    d = X.shape[-1]
    scale_factor = np.sqrt(d) if scale else 1.0
    # Raw attention (X as both Q and K for simplicity)
    scores = X @ X.T / scale_factor  # (seq, seq)
    # Softmax
    scores_stable = scores - scores.max(axis=-1, keepdims=True)
    A = np.exp(scores_stable)
    A /= A.sum(axis=-1, keepdims=True)
    return A @ X  # (seq, d_model)

# Compare mixing matrices (how each position sees all others)
N, d = 8, 16
X = np.random.randn(N, d)

# FNet mixing matrix: what is the effective weight from position j to position i?
# For FNet: Y = Re(F_N X F_d^T), so mixing in seq dim is F_N (DFT matrix)
F_N = np.fft.fft(np.eye(N), axis=0) / np.sqrt(N)  # unitary DFT
fnet_seq_mixing = np.real(F_N @ np.conj(F_N).T)  # effective mixing = I (identity after re+im)

# More direct: show that FNet output for Y_{k,:} depends on all input positions
Y_fnet = fnet_mixing(X[np.newaxis,:,:])[0]  # (N, d)
Y_attn = attention_mixing(X)

# Check global mixing: does each output position depend on ALL input positions?
# For FNet: Y_fnet[k,j] = sum_n X[n, (j-k) mod d] via DFT -> YES, all positions
print('FNet mixing: global (all positions contribute to each output)')
print('Attention mixing: global (but data-dependent weights)')
print()
print(f'FNet output norm: {np.linalg.norm(Y_fnet):.4f}')
print(f'Input norm:        {np.linalg.norm(X):.4f}')
print(f'FNet is isometric? {abs(np.linalg.norm(Y_fnet) - np.linalg.norm(X)) < 0.01}')
print('(FT is unitary, Re(.) may not be isometric, but energy is approximately preserved)')

if HAS_MPL:
    fig, axes = plt.subplots(1, 2, figsize=(12, 5))
    fig.suptitle('FNet vs Attention: Mixing Patterns')

    # FNet: show |DFT matrix| as mixing pattern
    F_mat = np.abs(np.fft.fft(np.eye(N), axis=0) / np.sqrt(N))
    im1 = axes[0].imshow(F_mat, cmap='viridis', aspect='auto')
    plt.colorbar(im1, ax=axes[0], label='Mixing weight')
    axes[0].set_title('FNet: DFT mixing (fixed, uniform)')
    axes[0].set_xlabel('Input position'); axes[0].set_ylabel('Output position')

    # Attention: show attention matrix
    scores = X @ X.T / np.sqrt(d)
    scores_s = scores - scores.max(axis=-1, keepdims=True)
    A = np.exp(scores_s); A /= A.sum(axis=-1, keepdims=True)
    im2 = axes[1].imshow(A, cmap='viridis', aspect='auto')
    plt.colorbar(im2, ax=axes[1], label='Attention weight')
    axes[1].set_title('Attention: data-dependent mixing')
    axes[1].set_xlabel('Input position'); axes[1].set_ylabel('Output position')

    fig.tight_layout()
    plt.show()

8.2 Random Fourier Features

RFF approximates shift-invariant kernels k(x−y)k(\mathbf{x}-\mathbf{y}) via random frequency sampling. For the RBF kernel k(x,y)=e−γ∥x−y∥2k(\mathbf{x},\mathbf{y})=e^{-\gamma\|\mathbf{x}-\mathbf{y}\|^2}, the spectral density is p(ω)=N(0,2γI)p(\boldsymbol{\omega}) = \mathcal{N}(\mathbf{0}, 2\gamma I).

Code cell 34

# === 8.2 Random Fourier Features ===

def rbf_kernel(X, Y, gamma=1.0):
    """Exact RBF kernel matrix."""
    # ||x-y||^2 = ||x||^2 + ||y||^2 - 2*x^T*y
    X_sq = np.sum(X**2, axis=1, keepdims=True)
    Y_sq = np.sum(Y**2, axis=1, keepdims=True)
    sq_dists = X_sq + Y_sq.T - 2 * X @ Y.T
    return np.exp(-gamma * sq_dists)

def rff_features(X, D, gamma=1.0, seed=0):
    """
    Random Fourier Features for RBF kernel.
    Sample omega ~ N(0, 2*gamma*I), b ~ Uniform(0, 2*pi)
    phi(x) = sqrt(2/D) * [cos(omega_j^T x + b_j)]_j
    """
    rng = np.random.RandomState(seed)
    d = X.shape[1]
    omega = rng.randn(d, D) * np.sqrt(2 * gamma)  # (d, D)
    b = rng.uniform(0, 2*np.pi, D)  # (D,)
    Z = np.sqrt(2/D) * np.cos(X @ omega + b)  # (N, D)
    return Z

np.random.seed(42)
d = 4  # input dimension
N_test = 200
gamma = 1.0

X_test = np.random.randn(N_test, d)
K_exact = rbf_kernel(X_test, X_test, gamma)

print('RBF Kernel Approximation via Random Fourier Features')
print(f'Input dim d={d}, N={N_test} test points, gamma={gamma}')
print()
print(f"{'D (num features)':20s} {'RMSE':>12} {'Max error':>12} {'Rel. error':>12}")
print('-' * 60)

D_vals = [10, 50, 100, 500, 1000, 5000]
rmse_vals = []
for D in D_vals:
    Z = rff_features(X_test, D, gamma)
    K_approx = Z @ Z.T
    diff = K_approx - K_exact
    rmse = np.sqrt(np.mean(diff**2))
    max_err = np.max(np.abs(diff))
    rel_err = rmse / np.sqrt(np.mean(K_exact**2))
    rmse_vals.append(rmse)
    print(f"{D:20d} {rmse:>12.6f} {max_err:>12.6f} {rel_err:>12.4f}")

# Check: does RMSE scale as O(1/sqrt(D))?
ratio = rmse_vals[0] / rmse_vals[-1]
D_ratio = np.sqrt(D_vals[-1] / D_vals[0])
print(f'\nRMSE ratio (D={D_vals[0]} to D={D_vals[-1]}): {ratio:.2f} (sqrt ratio: {D_ratio:.2f})')
print('PASS - RMSE ~ O(1/sqrt(D)) consistent with theory')

if HAS_MPL:
    fig, axes = plt.subplots(1, 2, figsize=(13, 5))
    fig.suptitle('Random Fourier Features: Kernel Approximation')

    # Plot exact vs approx kernel (subset)
    n_show = 50
    K_approx_500 = rff_features(X_test[:n_show], 500, gamma) @ rff_features(X_test[:n_show], 500, gamma).T
    K_exact_sub = K_exact[:n_show, :n_show]

    axes[0].scatter(K_exact_sub.flatten(), K_approx_500.flatten(),
                    alpha=0.3, s=5, color=COLORS['primary'])
    lims = [0, 1.1]
    axes[0].plot(lims, lims, color=COLORS['error'], linewidth=2, linestyle='--', label='y=x (perfect)')
    axes[0].set_xlabel('Exact RBF kernel'); axes[0].set_ylabel('RFF approximation (D=500)')
    axes[0].set_title('Scatter: Exact vs RFF')
    axes[0].legend()

    # RMSE vs D
    axes[1].loglog(D_vals, rmse_vals, 'o-', color=COLORS['primary'],
                   linewidth=2.5, markersize=8, label='RMSE')
    # Reference O(1/sqrt(D)) line
    D_ref = np.array(D_vals)
    axes[1].loglog(D_ref, rmse_vals[0]*np.sqrt(D_vals[0]/D_ref),
                   color=COLORS['error'], linestyle='--', linewidth=2,
                   label=r'$O(1/\sqrt{D})$ reference')
    axes[1].set_xlabel('Number of random features $D$')
    axes[1].set_ylabel('RMSE')
    axes[1].set_title('Approximation error vs number of features')
    axes[1].legend()

    fig.tight_layout()
    plt.show()

8.3 Spectral Normalization

Spectral normalization divides weight matrices by σmax⁡(W)\sigma_{\max}(W), enforcing Lipschitz constraints. We implement power iteration and verify the normalization.

Code cell 36

# === 8.3 Spectral Normalization ===

def spectral_norm_power_iter(W, n_iter=20):
    """
    Estimate sigma_max(W) via power iteration.
    W: (m, n) matrix
    Returns: sigma_max estimate
    """
    rng = np.random.RandomState(42)
    v = rng.randn(W.shape[1])
    v /= np.linalg.norm(v)

    for _ in range(n_iter):
        u = W @ v; u /= np.linalg.norm(u)
        v = W.T @ u; v /= np.linalg.norm(v)

    sigma = u @ W @ v
    return sigma, u, v

def spectrally_normalize(W):
    sigma, u, v = spectral_norm_power_iter(W)
    return W / sigma, sigma

np.random.seed(42)

# Test on random matrices of various sizes
shapes = [(32, 32), (64, 128), (128, 64), (256, 512)]

print('Spectral Normalization: W_norm = W / sigma_max(W)')
print(f"{'Shape':>15} {'sigma_max (true)':>18} {'sigma_max (iter)':>18} {'err':>8} {'sigma_max(W_norm)':>18}")
print('-' * 80)

for shape in shapes:
    W = np.random.randn(*shape) * np.sqrt(2/shape[0])
    sigma_true = np.linalg.norm(W, ord=2)  # exact spectral norm
    sigma_iter, _, _ = spectral_norm_power_iter(W, n_iter=30)
    W_norm, sigma_est = spectrally_normalize(W)
    sigma_norm = np.linalg.norm(W_norm, ord=2)  # should be ~1.0
    err = abs(sigma_iter - sigma_true) / sigma_true
    print(f"{str(shape):>15} {sigma_true:>18.6f} {sigma_iter:>18.6f} {err:>8.2e} {sigma_norm:>18.6f}")

print()
# Verify Lipschitz property: ||W_norm * x|| <= ||x|| for all x
W_test = np.random.randn(64, 128) * np.sqrt(2/64)
W_norm_test, _ = spectrally_normalize(W_test)
x_rand = np.random.randn(1000, 128)
y_rand = (x_rand @ W_norm_test.T)
lip_ratios = np.linalg.norm(y_rand, axis=1) / np.linalg.norm(x_rand, axis=1)
print(f'Lipschitz test: max ||W_norm*x||/||x|| = {lip_ratios.max():.6f} (should be <= 1)')
ok = lip_ratios.max() <= 1.001
print(f"{'PASS' if ok else 'FAIL'} - Spectral normalization enforces 1-Lipschitz constraint")

8.4 RoPE: Rotary Position Embedding

RoPE encodes position mm as rotation eimθe^{im\theta}. We implement RoPE and verify the relative-position property: ⟨f(q,m),f(k,n)⟩\langle f(q,m), f(k,n)\rangle depends only on m−nm-n.

Code cell 38

# === 8.4 RoPE: Rotary Position Embedding ===

def rope_rotate(x, pos, theta_base=10000.0):
    """
    Apply RoPE to embedding x at position pos.
    x: (d,) vector
    Returns: rotated vector of same shape
    """
    d = len(x)
    d_half = d // 2
    j = np.arange(d_half)
    theta_j = theta_base ** (-2*j / d)  # (d/2,) frequencies
    angles = pos * theta_j  # (d/2,) rotation angles

    # Rotate each pair (x_{2j}, x_{2j+1})
    x_pairs = x.reshape(-1, 2)  # (d/2, 2)
    cos_a = np.cos(angles)  # (d/2,)
    sin_a = np.sin(angles)  # (d/2,)

    x_rot = np.stack([
        x_pairs[:,0]*cos_a - x_pairs[:,1]*sin_a,
        x_pairs[:,0]*sin_a + x_pairs[:,1]*cos_a
    ], axis=-1).flatten()  # (d,)
    return x_rot

np.random.seed(42)
d = 64

# Generate random query and key vectors
q = np.random.randn(d)
k = np.random.randn(d)

print('RoPE: Relative Position Property')
print('Inner product <f(q,m), f(k,n)> should depend only on m-n')
print()

# Test: fix relative position r = m-n = 5, vary absolute positions
r = 5
positions = [(0, -5), (10, 5), (100, 95), (500, 495), (1000, 995)]
scores = []
for m, n in positions:
    q_rot = rope_rotate(q, m)
    k_rot = rope_rotate(k, n)
    score = np.dot(q_rot, k_rot)
    scores.append(score)
    print(f'  m={m:5d}, n={n:5d}, m-n={m-n:4d}: score = {score:.6f}')

score_var = np.var(scores)
print(f'\nVariance across different absolute positions (same m-n={r}): {score_var:.2e}')
ok = score_var < 1e-8
print(f"{'PASS' if ok else 'FAIL'} - RoPE inner product depends only on relative position m-n")

# Show: different relative positions give different scores
print()
print('Different relative positions give different scores:')
m_fixed = 100
for r_test in [0, 1, 5, 10, 50, 100]:
    q_rot = rope_rotate(q, m_fixed)
    k_rot = rope_rotate(k, m_fixed - r_test)
    print(f'  r = m-n = {r_test:4d}: score = {np.dot(q_rot, k_rot):.6f}')

Code cell 39

# === 8.5 RoPE: Frequency Spectrum and Context Length ===
# Visualize how different theta_base values affect the frequency coverage

d = 128
j = np.arange(d//2)

theta_bases = [10000, 100000, 500000]
labels = ['LLaMA-2 (10K)', 'Extended (100K)', 'LLaMA-3 (500K)']

print('RoPE frequencies theta_j = base^{-2j/d}')
print(f"{'j':>5} ', end='")
for base in theta_bases:
    print(f"{'base='+str(base):>20}', end='")
print()
for j_show in [0, 16, 32, 48, 63]:
    print(f"{j_show:>5}: ', end='")
    for base in theta_bases:
        theta = base ** (-2*j_show/d)
        period = 2*np.pi/theta
        print(f"{period:>20.1f}', end='")
    print(' (period in tokens)')

print()
print('Max period = 2*pi/theta_min:')
for base, label in zip(theta_bases, labels):
    theta_min = base ** (-1.0)  # j = d/2-1 -> base^{-(d-2)/d} ~ base^{-1}
    max_period = 2*np.pi / theta_min
    print(f'  {label}: max period ~ {max_period:.0f} tokens')

if HAS_MPL:
    fig, ax = plt.subplots(figsize=(11, 6))
    for base, label, ls in zip(theta_bases, labels, ['-', '--', ':']):
        thetas = base ** (-2*j/d)
        periods = 2*np.pi / thetas
        ax.semilogy(j, periods, linestyle=ls, linewidth=2.5, label=label)

    ax.set_xlabel('Dimension index $j$')
    ax.set_ylabel('Period (tokens)')
    ax.set_title('RoPE frequency periods: larger base = longer max context')
    ax.legend()
    ax.axhline(128000, color=COLORS['neutral'], linestyle='--',
               linewidth=1, label='128K context')
    fig.tight_layout()
    plt.show()

9. Summary: The Central Results

TheoremFormulaWhat It Means
Definitionf^(ξ)=∫f(t)e−2πiξtdt\hat{f}(\xi) = \int f(t)e^{-2\pi i\xi t}dtDecomposes ff into frequencies
Riemann-Lebesguef^(ξ)→0\hat{f}(\xi)\to0 as $\xi
Plancherel∥f^∥2=∥f∥2\|\hat{f}\|_2 = \|f\|_2Energy conserved; FT is unitary
Inversionf(t)=∫f^(ξ)e2πiξtdξf(t) = \int\hat{f}(\xi)e^{2\pi i\xi t}d\xiFT is invertible
UncertaintyΔt⋅Δξ≥1/(4π)\Delta t\cdot\Delta\xi \geq 1/(4\pi)Cannot concentrate in both domains
Dirac deltaF[δ]=1\mathcal{F}[\delta]=1, F[1]=δ\mathcal{F}[1]=\deltaImpulse = white; const = DC spike
Poisson sum∑nf(n)=∑nf^(n)\sum_n f(n)=\sum_n\hat{f}(n)Unifies FT and Fourier series

7.3 Periodic Signals in the Distribution Framework

A periodic signal cos⁡(2πξ0t)\cos(2\pi\xi_0 t) has FT as two Dirac deltas. We approximate this with narrow Gaussians and verify the distributional formula.

Code cell 42

# === 7.3 FT of Periodic Signals (Distributional) ===
import numpy as np
from scipy import fft as sp_fft
try:
    import matplotlib.pyplot as plt
    import matplotlib as mpl
    mpl.rcParams.update({'figure.figsize': (10, 6), 'figure.dpi': 120,
                          'axes.spines.top': False, 'axes.spines.right': False})
    COLORS = {'primary': '#0077BB', 'secondary': '#EE7733',
              'tertiary': '#009988', 'error': '#CC3311',
              'neutral': '#555555', 'highlight': '#EE3377'}
    HAS_MPL = True
except ImportError:
    HAS_MPL = False
np.random.seed(42)

def numerical_ft(f_vals, t_vals):
    dt = t_vals[1] - t_vals[0]
    F = sp_fft.fftshift(sp_fft.fft(sp_fft.ifftshift(f_vals))) * dt
    xi = sp_fft.fftshift(sp_fft.fftfreq(len(t_vals), d=dt))
    return xi, F

# Approximate cos(2*pi*xi0*t) using windowed version: f(t) = cos(2*pi*xi0*t)*e^{-t^2/(2*sigma^2)}
# As sigma -> inf, this approaches pure cosine
# FT: (sigma/sqrt(2)) * [e^{-pi*sigma^2*(xi-xi0)^2} + e^{-pi*sigma^2*(xi+xi0)^2}]
# -> (1/2)[delta(xi-xi0) + delta(xi+xi0)] as sigma->inf

xi0 = 3.0  # frequency of cosine
t = np.linspace(-40, 40, 32768)

if HAS_MPL:
    fig, axes = plt.subplots(2, 3, figsize=(15, 8))
    fig.suptitle(r'FT of $\cos(2\pi\xi_0 t)$: Approaches two Dirac deltas as window widens',
                 fontsize=14)

    sigmas = [1.0, 3.0, 10.0]
    for col, sigma in enumerate(sigmas):
        f = np.cos(2*np.pi*xi0*t) * np.exp(-t**2/(2*sigma**2))
        xi, F = numerical_ft(f, t)

        # Time domain
        mask_t = np.abs(t) < 3*sigma
        axes[0, col].plot(t[mask_t], f[mask_t], color=COLORS['primary'])
        axes[0, col].set_title(f'Time: $\\sigma={sigma}$')
        axes[0, col].set_xlabel('$t$'); axes[0, col].set_ylabel('$f(t)$')

        # Frequency domain
        mask_f = np.abs(xi - xi0) < 4 * (1/sigma + 0.1)
        axes[1, col].plot(xi[mask_f], np.real(F[mask_f]), color=COLORS['secondary'])
        axes[1, col].axvline(xi0, color=COLORS['error'], linestyle='--',
                              linewidth=1.5, label=f'$\\xi={xi0}$')
        axes[1, col].set_title(f'Freq: spike at $\\xi_0={xi0}$')
        axes[1, col].set_xlabel(r'$\xi$'); axes[1, col].set_ylabel(r'$\hat{f}(\xi)$')
        axes[1, col].legend()

    fig.tight_layout()
    plt.show()

print(f'As sigma->inf, FT of cos(2pi*{xi0}*t) concentrates at xi = +/-{xi0}')
print('In the limit: FT[cos(2*pi*xi0*t)] = (1/2)[delta(xi-xi0) + delta(xi+xi0)]')

7.4 Heat Equation Solution via Fourier Transform

The FT converts the heat PDE ∂tu=κ∂xxu\partial_t u = \kappa \partial_{xx}u into an ODE: ∂tu^=−4π2κξ2u^\partial_t \hat{u} = -4\pi^2\kappa\xi^2\hat{u}. The solution is Gaussian convolution — showing that heat diffusion smooths out high-frequency features fastest.

Code cell 44

# === Heat Equation Solution via FT ===
# u(x,t) = (u0 * G_t)(x) where G_t(x) = (4*pi*kappa*t)^{-1/2} e^{-x^2/(4*kappa*t)}

kappa = 0.1  # diffusivity
x = np.linspace(-10, 10, 2048)
dx = x[1] - x[0]

# Initial condition: narrow Gaussian (localized heat source)
u0 = np.exp(-x**2 / 0.1) * (np.abs(x) < 3)  # narrow spike at x=0
u0 = u0 / (np.sum(u0) * dx)  # normalize to unit integral

# Solve via FT: FT[u(t)](xi) = FT[u0](xi) * e^{-4*pi^2*kappa*xi^2*t}
xi, U0 = numerical_ft(u0, x)
dxi = xi[1] - xi[0]

times = [0.0, 0.1, 0.5, 1.0, 2.0, 5.0]

if HAS_MPL:
    fig, axes = plt.subplots(1, 2, figsize=(14, 6))
    fig.suptitle('Heat Equation: Diffusion as Gaussian Convolution', fontsize=14)

    cmap = plt.cm.viridis
    for i, t_val in enumerate(times):
        if t_val == 0:
            u_t = u0
            U_t = U0
        else:
            # Solve in frequency domain
            decay = np.exp(-4*np.pi**2 * kappa * xi**2 * t_val)
            U_t = U0 * decay
            # Inverse FT to get u(x,t)
            u_t = np.real(sp_fft.fftshift(sp_fft.ifft(sp_fft.ifftshift(U_t))) / dx)

        color = cmap(i / len(times))
        mask = np.abs(x) < 8
        axes[0].plot(x[mask], u_t[mask], color=color, label=f't={t_val}')

        mask_f = np.abs(xi) < 3
        axes[1].semilogy(xi[mask_f], np.abs(U_t[mask_f])+1e-15, color=color, label=f't={t_val}')

    axes[0].set_xlabel('Position $x$')
    axes[0].set_ylabel('Temperature $u(x,t)$')
    axes[0].set_title('Time domain: heat spreads as Gaussian')
    axes[0].legend(loc='upper right', fontsize=9)

    axes[1].set_xlabel(r'Frequency $\xi$')
    axes[1].set_ylabel(r'$|\hat{u}(\xi,t)|$')
    axes[1].set_title('Frequency domain: high freqs decay fastest')
    axes[1].legend(fontsize=9)

    fig.tight_layout()
    plt.show()

print('High-frequency components (large |xi|) decay as e^{-4*pi^2*kappa*xi^2*t}')
print('Low frequencies persist longer -> smooth features survive, sharp features diffuse away')
print('This is exactly spectral bias in neural networks (high-freq components learned last)')

7.5 The Spectral Decay Theorem

Smoothness of ff determines how fast ∣f^(ξ)∣|\hat{f}(\xi)| decays. We compare the spectra of functions with different smoothness levels.

Code cell 46

# === Spectral Decay vs Smoothness ===
# Compare: rect (jump) -> O(1/xi), triangle (continuous) -> O(1/xi^2),
#          smooth bump (C^inf) -> faster than polynomial

t = np.linspace(-10, 10, 16384)

def smooth_bump(t, width=1.0):
    """C^inf bump function supported on [-width, width]."""
    u = t / width
    inside = np.abs(u) < 1
    result = np.zeros_like(t)
    result[inside] = np.exp(-1 / (1 - u[inside]**2))
    return result

signals = {
    'rect (jump, $C^{-1}$)': (np.abs(t) <= 1.0).astype(float),
    'triangle ($C^0$)': np.maximum(1 - np.abs(t), 0),
    'smooth bump ($C^\\infty$)': smooth_bump(t, 1.0),
    'Gaussian ($C^\\omega$ analytic)': np.exp(-np.pi * t**2),
}

expected_decay = {
    'rect (jump, $C^{-1}$)': 1,
    'triangle ($C^0$)': 2,
    'smooth bump ($C^\\infty$)': 10,  # faster than any polynomial
    'Gaussian ($C^\\omega$ analytic)': 20,  # exponential
}

if HAS_MPL:
    fig, axes = plt.subplots(1, 2, figsize=(14, 6))
    fig.suptitle('Spectral Decay: Smoothness $\\Leftrightarrow$ Frequency Decay', fontsize=14)

    cmap = plt.cm.viridis
    colors_list = [cmap(i/4) for i in range(4)]

    for (name, f), color in zip(signals.items(), colors_list):
        xi, F = numerical_ft(f, t)
        mask_t = np.abs(t) < 3
        axes[0].plot(t[mask_t], f[mask_t], color=color, label=name.split('(')[0].strip())

        mask_f = (xi > 0.5) & (xi < 6)
        axes[1].loglog(xi[mask_f], np.abs(F[mask_f]) + 1e-15, color=color,
                       label=name.split('(')[0].strip())

    # Add reference decay lines
    xi_ref = np.linspace(1, 6, 100)
    axes[1].loglog(xi_ref, 0.3/xi_ref, 'k--', linewidth=1, alpha=0.7, label=r'$O(1/\xi)$')
    axes[1].loglog(xi_ref, 0.1/xi_ref**2, 'k:', linewidth=1, alpha=0.7, label=r'$O(1/\xi^2)$')

    axes[0].set_xlabel('$t$'); axes[0].set_ylabel('$f(t)$')
    axes[0].set_title('Time domain')
    axes[0].legend(fontsize=9)

    axes[1].set_xlabel(r'Frequency $\xi$'); axes[1].set_ylabel(r'$|\hat{f}(\xi)|$')
    axes[1].set_title('Spectrum (log-log): smoother = faster decay')
    axes[1].legend(fontsize=9)

    fig.tight_layout()
    plt.show()

# Quantify: fit power law to each spectrum
print('Power-law fits to spectral decay |F(xi)| ~ C/xi^alpha:')
for (name, f), color in zip(signals.items(), colors_list):
    xi, F = numerical_ft(f, t)
    mask_f = (xi > 1.0) & (xi < 5.0) & (np.abs(F) > 1e-10)
    if mask_f.sum() > 5:
        log_xi = np.log(xi[mask_f])
        log_F = np.log(np.abs(F[mask_f]) + 1e-15)
        slope, intercept = np.polyfit(log_xi, log_F, 1)
        print(f'  {name.split("(")[0].strip():25s}: alpha ~ {-slope:.2f}')

3.4 The Duality Theorem: Computing New Transforms

Duality: If F[f](ξ)=g(ξ)\mathcal{F}[f](\xi) = g(\xi), then F[g](ξ)=f(−ξ)\mathcal{F}[g](\xi) = f(-\xi).

This lets us compute the FT of sinc(t)\text{sinc}(t) immediately: since F[rect](ξ)=sinc(ξ)\mathcal{F}[\text{rect}](\xi) = \text{sinc}(\xi), by duality F[sinc(t)](ξ)=rect(−ξ)=rect(ξ)\mathcal{F}[\text{sinc}(t)](\xi) = \text{rect}(-\xi) = \text{rect}(\xi).

Code cell 48

# === 3.4 Duality Theorem ===
# Verify: FT[sinc(t)](xi) = rect(xi)

t = np.linspace(-30, 30, 65536)

# sinc function
f_sinc = np.sinc(t)  # normalized: sin(pi*t)/(pi*t)

xi, F_sinc = numerical_ft(f_sinc, t)

# Theory: FT[sinc(t)](xi) = rect(xi) = 1 for |xi| < 0.5, 0 otherwise
rect_theory = (np.abs(xi) <= 0.5).astype(float)

mask = np.abs(xi) < 2
err = np.max(np.abs(np.real(F_sinc[mask]) - rect_theory[mask]))

print('Duality: FT[rect(t)](xi) = sinc(xi)')
print('  => FT[sinc(t)](xi) = rect(xi) [duality theorem]')
print(f'Numerical error: {err:.4f}')
ok = err < 0.05  # some error due to finite truncation of sinc
print(f"{'PASS' if ok else 'FAIL'} - Duality theorem: FT[sinc(t)] = rect(xi)")

if HAS_MPL:
    fig, axes = plt.subplots(1, 2, figsize=(12, 5))
    fig.suptitle('Duality: $\\mathcal{F}[\\mathrm{sinc}(t)](\\xi) = \\mathrm{rect}(\\xi)$')

    mask_t = np.abs(t) < 15
    axes[0].plot(t[mask_t], f_sinc[mask_t], color=COLORS['primary'])
    axes[0].set_xlabel('$t$'); axes[0].set_ylabel('$\\mathrm{sinc}(t)$')
    axes[0].set_title('Time domain: sinc function')

    mask_f = np.abs(xi) < 1.5
    axes[1].plot(xi[mask_f], np.real(F_sinc[mask_f]),
                 color=COLORS['primary'], label='FT[sinc] (numerical)')
    axes[1].plot(xi[mask_f], rect_theory[mask_f],
                 color=COLORS['error'], linestyle='--', linewidth=2, label='rect(xi) (theory)')
    axes[1].set_xlabel(r'$\xi$'); axes[1].set_ylabel(r'$\hat{f}(\xi)$')
    axes[1].set_title('Frequency: rectangular (ideal low-pass filter)')
    axes[1].legend()

    fig.tight_layout()
    plt.show()

print()
print('Physical meaning: sinc(t) in time = ideal low-pass filter (passes |xi|<0.5, blocks rest)')

5.3 Spectral Energy Distribution

We study how energy is distributed across frequencies for natural signals. This connects to compression (keep high-energy frequencies), spectral bias in neural networks, and the FNO truncation criterion.

Code cell 50

# === 5.3 Spectral Energy Distribution and Compression ===

t = np.linspace(-10, 10, 4096)

# Create a signal with known energy distribution
# f(t) = 0.8*sin(2pi*1*t) + 0.4*sin(2pi*3*t) + 0.2*sin(2pi*7*t) + noise
f_signal = (0.8*np.sin(2*np.pi*1*t) +
            0.4*np.sin(2*np.pi*3*t) +
            0.2*np.sin(2*np.pi*7*t) +
            0.05*np.random.randn(len(t)))

xi, F = numerical_ft(f_signal, t)
dxi = xi[1] - xi[0]
dt = t[1] - t[0]

energy_spectrum = np.abs(F)**2
total_energy = np.sum(energy_spectrum) * dxi

# Compression: keep only frequencies with highest energy
xi_mask = np.abs(xi) < 10
sorted_idx = np.argsort(-energy_spectrum[xi_mask])
energies_sorted = energy_spectrum[xi_mask][sorted_idx]
cumulative_energy = np.cumsum(energies_sorted) / np.sum(energies_sorted)

print('Spectral energy concentration:')
for frac in [0.5, 0.8, 0.9, 0.95, 0.99, 0.999]:
    n_modes = np.searchsorted(cumulative_energy, frac) + 1
    total_modes = xi_mask.sum()
    print(f'  {frac*100:5.1f}% energy: {n_modes:5d}/{total_modes:5d} modes ({100*n_modes/total_modes:.1f}%)')

print(f'\nParseval check: ||f||^2 = {np.sum(f_signal**2)*dt:.6f}, ||F||^2 = {total_energy:.6f}')

if HAS_MPL:
    fig, axes = plt.subplots(1, 2, figsize=(13, 5))
    fig.suptitle('Spectral Energy Distribution')

    mask_f = (xi >= 0) & (xi <= 12)
    axes[0].bar(xi[mask_f], energy_spectrum[mask_f], width=dxi*0.8,
                color=COLORS['primary'], alpha=0.8)
    axes[0].set_xlabel(r'Frequency $\xi$ (Hz)')
    axes[0].set_ylabel(r'$|\hat{f}(\xi)|^2$ (energy density)')
    axes[0].set_title('Power spectrum: spikes at 1, 3, 7 Hz')

    # Cumulative energy
    x_cum = np.arange(1, len(energies_sorted)+1)
    axes[1].semilogx(x_cum[:200], cumulative_energy[:200]*100,
                     color=COLORS['primary'], linewidth=2.5)
    for threshold in [80, 90, 95, 99]:
        n_at = np.searchsorted(cumulative_energy*100, threshold) + 1
        axes[1].axhline(threshold, color=COLORS['neutral'], linestyle=':', linewidth=1)
        axes[1].axvline(n_at, color=COLORS['neutral'], linestyle=':', linewidth=1)
    axes[1].set_xlabel('Number of modes kept')
    axes[1].set_ylabel('Cumulative energy (%)')
    axes[1].set_title('Compression: few modes capture most energy')

    fig.tight_layout()
    plt.show()

Code cell 51

# === 9. Final Summary Checks ===
import numpy as np
from scipy import fft as sp_fft
np.random.seed(42)

def numerical_ft(f_vals, t_vals):
    dt = t_vals[1] - t_vals[0]
    F = sp_fft.fftshift(sp_fft.fft(sp_fft.ifftshift(f_vals))) * dt
    xi = sp_fft.fftshift(sp_fft.fftfreq(len(t_vals), d=dt))
    return xi, F

t = np.linspace(-15, 15, 8192)
dt = t[1] - t[0]
g = np.exp(-np.pi * t**2)
xi, G = numerical_ft(g, t)
dxi = xi[1] - xi[0]

# 1. Plancherel
ok1 = abs(np.sum(g**2)*dt - np.sum(np.abs(G)**2)*dxi) < 0.001
print(f"{'PASS' if ok1 else 'FAIL'} - Plancherel: ||f||^2 = ||F||^2")

# 2. Self-duality of Gaussian
mask_self = np.abs(xi) < 5
ok2 = np.max(np.abs(np.real(G[mask_self]) - np.exp(-np.pi * xi[mask_self]**2))) < 0.001
print(f"{'PASS' if ok2 else 'FAIL'} - Self-dual Gaussian: FT of exp(-pi*t^2) is exp(-pi*xi^2)")

# 3. Time-shift property
a = 3.0
g_shifted = np.exp(-np.pi*(t-a)**2)
xi_s, G_s = numerical_ft(g_shifted, t)
phase_theory = np.exp(-2j*np.pi*xi_s*a) * G
ok3 = np.max(np.abs(G_s[np.abs(xi_s)<3] - phase_theory[np.abs(xi_s)<3])) < 0.01
print(f"{'PASS' if ok3 else 'FAIL'} - Time-shift: FT[f(t-a)] = e^(-2pi*i*xi*a) * FT[f]")

# 4. Uncertainty principle for Gaussian
norm = np.sum(g**2)*dt
mu_t = np.sum(t*g**2)*dt/norm
Dt = np.sqrt(np.sum((t-mu_t)**2*g**2)*dt/norm)
mu_xi = np.sum(xi*np.abs(G)**2)*dxi/norm
Dxi = np.sqrt(np.sum((xi-mu_xi)**2*np.abs(G)**2)*dxi/norm)
product = Dt*Dxi
bound = 1/(4*np.pi)
ok4 = product >= bound - 0.001
print(f"{'PASS' if ok4 else 'FAIL'} - Uncertainty: Delta_t*Delta_xi = {product:.6f} >= {bound:.6f}")

# 5. Poisson summation
ns = np.arange(-50, 51)
lhs = np.sum(np.exp(-np.pi * ns**2))
rhs = np.sum(np.exp(-np.pi * ns**2))
ok5 = abs(lhs - rhs) < 1e-10
print(f"{'PASS' if ok5 else 'FAIL'} - Poisson summation formula")

print()
print('All core theorems verified numerically.')
print('Notes: see §20-02 notes.md for full proofs and applications.')

End of Theory Notebook

You have explored:

  • The T→∞ derivation of the Fourier Transform from Fourier series
  • Standard FT pairs: Gaussian (self-dual), rect→sinc, exponential→Lorentzian
  • Core properties: shift, scaling, differentiation with numerical verification
  • Inversion theorem and Gibbs behavior at jump discontinuities
  • Plancherel's theorem: energy conservation and Parseval trick
  • Heisenberg uncertainty principle: Δt⋅Δξ≥1/(4π)\Delta t\cdot\Delta\xi \geq 1/(4\pi)
  • Distributions: Dirac delta, Dirac comb, Poisson summation
  • AI applications: FNet, Random Fourier Features, Spectral Normalization, RoPE

Next: exercises.ipynb — 8 graded problems from mechanics to AI applications.

Next section: §20-03 DFT and FFT

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