Enhanced Mollifier Functions Library

🔬 Explore Mollifiers and Function Smoothing

Mollifiers are smooth functions with special properties used to create smooth approximations of non-smooth functions. They are fundamental tools in mathematical analysis, particularly in the theory of partial differential equations and distribution theory.

A mollifier is typically a smooth function φ with compact support that satisfies:

• φ ∈ C^∞ (infinitely differentiable)
• φ has compact support (identically zero outside some interval)
• ∫φ(x)dx = 1 (normalized to have unit integral)
• φ_ε(x) = (1/ε)φ(x/ε) (scaled version with parameter ε)

The convolution of a function f with a mollifier φ_ε produces a smooth approximation of f:
f_ε(x) = (f ∗ φ_ε)(x) = ∫f(y)φ_ε(x-y)dy
As ε → 0, f_ε converges to f in various senses depending on the properties of f.

Learn more about mollifiers on Wikipedia.

Applications:

• Regularization of non-smooth functions
• Proofs of density theorems in function spaces
• Construction of smooth partitions of unity
• Numerical analysis and signal processing
• Regularization of ill-posed problems