In mathematics, a mollifier (also known as an approximation to the identity) is a smooth function used to create sequences of smooth functions that approximate nonsmooth or generalized functions through the operation of convolution. Intuitively, given a function with sharp or irregular features, convolving it with a mollifier "mollifies" those features—smoothing them out while remaining close to the original function in an appropriate sense.
Mollifiers are also referred to as Friedrichs mollifiers, after the mathematician Kurt Otto Friedrichs, who introduced them in a 1944 paper ("The identity of weak and strong extensions of differential operators," Transactions of the American Mathematical Society). However, the Russian mathematician Sergei Sobolev had already employed closely related integral operators in his 1938 paper proving the Sobolev embedding theorem; Friedrichs himself acknowledged Sobolev's prior work, stating that "these mollifiers were introduced by Sobolev and the author."
Historical origin of the name
The term "mollifier" has a distinctive origin. According to Peter Lax's commentary in Friedrichs' Selecta, Friedrichs consulted his colleague Donald Flanders—a modern-day puritan nicknamed "Moll" after Moll Flanders—for advice on naming the smoothing operator he was using. Flanders suggested "mollifier," a pun combining his own nickname with the verb "to mollify," meaning "to smooth over." Notably, in Friedrichs' original usage, "mollifier" referred to the integral (convolution) operator itself, whereas in modern usage the term has been transferred to its kernel function.
Definition
Let φ be a smooth function on ℝⁿ (n ≥ 1), and define
φε(x) := ε−n φ(x/ε) for ε > 0.
Then φ is a mollifier if it satisfies:
- It is compactly supported;
- ∫ℝⁿ φ(x) dx = 1 (normalization);
- limε→0 φε(x) = δ(x), where δ is the Dirac delta function, with the limit understood in the space of Schwartz distributions.
Additional conditions may be imposed. If φ(x) ≥ 0 for all x, it is called a positive mollifier; if φ(x) = μ(|x|) for some smooth function μ, it is called a symmetric mollifier.
Concrete example
A standard example is the bump function
φ(x) = e−1/(1−|x|²) / In if |x| < 1, and 0 if |x| ≥ 1,
where In is a normalization constant. This function is infinitely differentiable, compactly supported on the unit ball, and defines a positive and symmetric mollifier.
Key properties
The behavior of mollifiers is governed by the operation of convolution:
- Smoothing property: For any distribution T, the family of convolutions Tε = T ∗ φε consists of smooth functions.
- Approximation of identity: As ε → 0, Tε converges to T in the appropriate space of distributions.
- Support of convolution: supp(T ∗ φε) ⊂ supp(T) + supp(φε), where "+" denotes Minkowski addition.
Applications
Mollifiers are fundamental tools in analysis and partial differential equations. They are used to:
- Prove that properties valid for smooth functions extend to nonsmooth situations;
- Define the multiplication (product) of distributions in theories of generalized functions;
- Prove the identity of "weak" and "strong" extensions of differential operators (the original purpose in Friedrichs' 1944 paper);
- Construct smooth cutoff functions and smooth partitions of unity, by convolving characteristic functions of sets with mollifiers.
These constructions are central to the theory of distributions and to the modern theory of partial differential equations.
