Dirac Approximate Identity

Dirac Approximate Identity Theory

The Dirac delta function is not an ordinary function but a measure or distribution. We can approximate it with a family of functions, called an approximate identity.

Consider the family \delta_ε(x) = (ε)/(π(x² + ε²)) for ε > 0. As ε → 0, this concentrates unit mass at the origin.

Weak convergence controls continuous test functions, not arbitrary indicator sets with boundary mass. The total integral is 1, and the cosine-test integral is exp(-ε|k|).

Experiment

Explore the approximate identity in the interactive module.

Select Width epsilon to see the density concentrate and observe Peak density become Unbounded as epsilon approaches 0.
Select Probe x to see the Point density decrease away from the origin.
Select Closed window radius to see how the total mass is affected.
Select Cosine frequency to see how the test integral is affected.

Worked Example

Let's calculate the values for a specific case and observe the results. Set Width epsilon to 0.5, Probe x to 0, Closed window radius to 0.5, and Cosine frequency to 1.

Select Width epsilon: 0.5
Select Probe x: 0
Select Closed window radius: 0.5
Select Cosine frequency: 1
Point density at x=0: 0.5/(π(0² + 0.5²)) = 0.5/(0.25π) = 2/(π) ≈ 0.6366197723675814
Closed window mass for R=0.5: 2/(π) arctan(0.5/0.5) = 2/(π) · (π)/4 = 0.5
Cosine test integral for k=1: exp(-0.5 · 1) ≈ 0.6065306597126334

Limitations

The visualization is limited to a finite domain and a specific approximate identity family (the Poisson kernel / Cauchy distribution). Other families like Gaussian exist.

References

For more rigorous details on delta sequences and approximate identities, refer to the following resource.

NIST DLMF: Delta Sequence

Dirac Approximate Identity