Understanding Delta Function Approximations: Sinc-Based Approximation (Fourier Kernel)

Delta-Convergent Sequences — Refined Blog with SageMath Symbolics, Physics Insights, and Cleaner Code

In the previous blog, we understood the Lorentzian Delta Sequence (Cauchy Kernel) and Gaussian Approximation (Heat Kernel) . Let's take another one step and explore the Sinc-Based Approximation (Fourier Kernel).

Why Study These Approximations?

Delta functions are central in many fields:

• Signal Processing: Ideal impulse, filter response
• Physics: Point charges/masses, Green's functions
• Spectral Theory: Lorentzian profiles in resonance
• Diffusion Models: Gaussians arise from the heat equation
• Numerics: Regularizing singular integrals

Each kernel has a story to tell.

Sinc-Based Approximation (Fourier Kernel)

Formula (with x = 0 defined): \[ f_{\nu}(x) = \begin{cases} \frac{\sin(\nu x)}{\pi x}, & x \neq 0 \\ \frac{\nu}{\pi}, & x = 0 \end{cases} \]

• SOscillatory, from Fourier analysis
• Not always positive
• Still integrates to 1

#Define the Function

var('x nu')
f_sinc(x, nu) = (1/pi) * (sin(nu * x) / x)
f_sinc(x, nu)

#Symbolic Integration Check
var('xi')
assume(nu > 0)  # Ensure nu is positive
integral(f_sinc(xi, nu), xi, -oo, oo).simplify_full()

#Limit at ( x = 0 )
limit(f_sinc(x, nu), x=0)

#Alternative Approach: Numerical Evaluation
x_vals = [0.1, 0.01, 0.001, 0.0001]
[f_sinc(x, 30).n() for x in x_vals]

#Integral Test (Distributional Behavior)
var('a b')
assume(a < 0, b > 0)  # Ensure a < 0 < b to match delta behavior
integral(f_sinc(xi, nu), xi, a, b).simplify_full()

#Numerical Verification

import numpy as np
import matplotlib.pyplot as plt
import sage.all as sage

def sinc_integral(nu, a=-1, b=1):
    from scipy.integrate import quad
    return quad(lambda x: np.sin(nu*x) / (np.pi*x), a, b)[0]

# Test for different ν values
nu_values = np.linspace(10, 100, 50)
integral_values = [sinc_integral(nu) for nu in nu_values]

# Plotting
plt.figure(figsize=(8, 5))
plt.plot(nu_values, integral_values, marker='o', linestyle='-', color='blue')
plt.axhline(y=1, color='r', linestyle='--', label="Expected Limit (1)")
plt.xlabel(r"$\nu$")
plt.ylabel(r"Integral Value")
plt.title("Numerical Verification: Sinc Integral Convergence")
plt.legend()
plt.grid(True)
plt.show()

#Plot the Sinc Function

p1 = plot(f_sinc(x, 10), (x, -5, 5), color='red', legend_label="ν=10") + \
     plot(f_sinc(x, 30), (x, -5, 5), color='blue', legend_label="ν=30") + \
     plot(f_sinc(x, 100), (x, -5, 5), color='green', legend_label="ν=100")

p1.show(title="Sinc Approximation to δ(x)", ymin=-1, ymax=3)

#First & Second Derivative Computation

f_sinc_prime(x, nu) = diff(f_sinc(x, nu), x)
f_sinc_double_prime(x, nu) = diff(f_sinc_prime(x, nu), x)

f_sinc_prime(x, nu), f_sinc_double_prime(x, nu)

#Plot the Derivatives
p1 = plot(f_sinc_prime(x, 10), (x, -5, 5), color='red', legend_label="ν=10") + \
     plot(f_sinc_prime(x, 30), (x, -5, 5), color='blue', legend_label="ν=30") + \
     plot(f_sinc_prime(x, 100), (x, -5, 5), color='green', legend_label="ν=100")

p1.show(title="First Derivative of Sinc Approximation")

p2 = plot(f_sinc_double_prime(x, 10), (x, -5, 5), color='red', legend_label="ν=10") + \
     plot(f_sinc_double_prime(x, 30), (x, -5, 5), color='blue', legend_label="ν=30") + \
     plot(f_sinc_double_prime(x, 100), (x, -5, 5), color='green', legend_label="ν=100")

p2.show(title="Second Derivative of Sinc Approximation")

p2.show(title="Second Derivative of Gaussian Delta Approximation")

#Integration of the Sinc Sequence
# Compute symbolic integral over a finite range (-a to b)
var('a b')
assume(a < 0, b > 0)
integral(f_sinc(x, nu), x, a, b).simplify_full()

#Plotting the Integrated Sequences
p1 = plot(integral(f_sinc(x, 30), x, -5, 5), (x, -5, 5), color='red', legend_label="Sinc")
p1.show(title="Integrated Delta Approximations")

💡 Try It Yourself! Now You can copy and paste directly into here Run SageMath Code Here

Physics Note
From Fourier theory and sampling, basis of Shannon’s sampling theorem.