Test Functions and Schwartz Space Explained – 𝓚 ⊂ 𝓢, with SageMath & SymPy!

Function Space Clubs: Who Belongs Where?

In mathematical analysis, we often group functions into "spaces" based on their behavior—how smooth they are, how fast they vanish at infinity, and how well-behaved they are under transformations like Fourier analysis.

𝓚 – The Compact Support Crew

• Infinitely differentiable
• Supported within a finite interval
• Vanishes outside a compact set

Think: A smooth “bump” function that’s perfectly zero beyond a certain region.

SageMath/SymPy Example:

      
import sympy

x = sympy.Symbol('x')

def k_function(x_val):
    if -1 <= x_val <= 1:
        return sympy.exp(-1 / (1 - x_val**2))  # Smooth bump
    else:
        return 0
	
    

𝓢 – The Schwartz Space Superstars

• Infinitely differentiable
• All derivatives decay faster than any polynomial as ∣𝑥∣→∞ \[ \phi(x) = e^{-x^2} \in \mathcal{S} \]

Gaussian functions are prime examples. They vanish "faster than fast"—ideal for signal processing, quantum mechanics, and PDEs.

𝓢′ – The Dual Space (Tempered Distributions)

• Contains linear functionals acting on 𝑆
• Includes generalized functions like Dirac delta, and integration functionals

Think: A smooth “bump” function that’s perfectly zero beyond a certain region.

SageMath/SymPy Example:

      
# Example functional acting on 𝓢
T(φ) = ∫ g(x) * φ(x) dx
# g can grow polynomially
	
    

Function Space Comparison

Function Space	Properties	Decay Behavior
𝓚 (Compact Support)	Infinitely differentiable, finite support	Zero outside a bounded interval
𝓢 (Schwartz Space)	Infinitely differentiable, all derivatives decay fast	Faster than any polynomial as \( |x| \to \infty \)
𝓢′ (Dual of 𝓢)	Linear functionals on 𝓢	Includes Dirac delta, polynomial-growth integrals

Visualizing the Gaussian Decay in 𝓢

Let’s plot \[ \phi(x) = e^{-x^2} \],the prototypical Schwartz function.

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

x = sympy.Symbol('x')
s_function = sympy.exp(-x**2)

f_lambdified = sympy.lambdify(x, s_function, 'numpy')

x_vals = np.linspace(-10, 10, 200)
y_vals = f_lambdified(x_vals)

plt.plot(x_vals, y_vals, label="e^(-x²)", color='blue')
plt.xlabel("x")
plt.ylabel("φ(x)")
plt.title("Decay Behavior of a Function in 𝓢")
plt.legend()
plt.grid(True)
plt.show()
	
    

Interactive Plot: This SageMathCell lets you explore how the Gaussian function φ(x) = e^(–x²) behaves. It's a classic Schwartz function with rapid decay.

💡 Explore the Gaussian Decay Behavior:
Copy the code snippet from this post and paste it into the live SageMath cell here:

👉 Run SageMath Code Here

Plot Description: A symmetric, bell-shaped curve centered at x = 0. The function rapidly approaches zero as x moves away from the center, illustrating Schwartz space decay.

𝓚 ⊂ 𝓢 – And 𝓚 is Dense in 𝓢

• Every 𝓚 function is also in 𝓢 (compact support → rapid decay).
• Density: You can approximate any Schwartz function using smooth compact-supported functions.

Example Approximation:

\[ \phi_v(x) = \phi(x) \cdot \eta_v(x) \] Where:

• \( \eta_v(x)\) is a smooth function: \( \eta_v(x)=1 \) on [-v,v],decaying smoothly outside
• As \( v \to \infty, \quad \phi_v \to \phi \) in 𝓢

Real-World Applications of Schwartz Space

• Fourier Analysis: F(S)⊂S
• Quantum Mechanics: Gaussians model wave packets
• PDE Theory: Weak solutions often live in 𝓢′

Fun Fact: The Fourier transform of \( \phi(x) = e^{-x^2} \) is another Gaussian—this symmetry makes it beloved in physics and math.

Try It Yourself!

Example-1

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

x = sympy.Symbol('x')
gaussian = sympy.exp(-x**2)

def smooth_cutoff(val, center, width):
    if -width/2 <= val - center <= width/2:
        return 1
    elif -width <= val - center < -width/2:
        return sympy.exp(-1 / (1 - ((val - center + 0.75*width) / (0.25*width))**2))
    elif width/2 < val - center <= width:
        return sympy.exp(-1 / (1 - ((val - center - 0.75*width) / (0.25*width))**2))
    else:
        return 0

widths = [1, 2, 3, 5]
plt.figure(figsize=(12, 8))
plt.plot(np.linspace(-6, 6, 400), [float(gaussian.subs(x, val)) for val in np.linspace(-6, 6, 400)], label='Gaussian', linestyle='--')

for w in widths:
    cutoff_sym = smooth_cutoff(x, 0, w)
    approximated_vals = [float((gaussian * cutoff_sym).subs(x, val)) for val in np.linspace(-6, 6, 400)]
    plt.plot(np.linspace(-6, 6, 400), approximated_vals, label=f'Width = {w}')

plt.title('Approximating Gaussian with Smoothly Cut-off Functions')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.legend()
plt.grid(True)
plt.ylim(-0.1, 1.1)
plt.show()
	
    

💡 Approximating Gaussian with Smoothly Cut-off Functions
Copy the code snippet from this post and paste it into the live SageMath cell here:

👉 Run SageMath Code Here

• Modify the Gaussian to \( e^{-x^4} \) or \( e^{-x^2}cos(x) \)
• Build your own compact-supported bump
• Test convergence using visual plots
• Explore the effect of Fourier transforms on these functions

Call to Action

Try implementing your own test function spaces in SageMath or SymPy! Explore convergence, visual decay, or even Fourier transforms.
💬 Comment below with your discoveries or any cool insights you uncover!
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