Distributions 13 | Convolution

TL;DR

For f, g ∈ L1(R^n), convolution is defined by (f * g)(x) = ∫ f(x − y) g(y) dy and satisfies ||f * g||_1 ≤ ||f||_1 ||g||_1.

Briefing Cornell Notes

Briefing

Convolution, first defined for ordinary integrable functions, can be extended to distributions by shifting the definition onto test functions and using duality—turning the star operation into a well-defined “multiplication” on distributions. For functions f, g in L1(R^n), convolution is built by mirroring one function and integrating the product over all shifts: (f * g)(x) = ∫ f(x − y) g(y) dy. This construction produces a new integrable function, and it satisfies a key norm estimate: ||f * g||_1 ≤ ||f||_1 ||g||_1. In other words, convolution behaves like multiplication inside the L1 algebra, complete with distributive rules.

The real challenge is making this operation meaningful when one or both inputs are distributions rather than honest functions. The approach starts with a locally integrable function f (a “regular distribution”) and a test function φ (smooth with compact support). The convolution is then defined indirectly through how it acts under integration against test functions: the pairing ⟨φ * f, ψ⟩ is rewritten by expanding the convolution inside the integral. By swapping the order of integration—justified because the test function has compact support—the expression collapses into a new pairing involving f and a convolution of test functions.

A crucial technical step is the “reflected” test function, written as φ̌, meaning φ̌(z) = φ(−z). This reflection appears because the convolution formula contains x − y, and changing the order of variables forces a sign flip. After this rearrangement, the pairing for regular distributions becomes ⟨φ * f, ψ⟩ = ⟨f, φ̌ * ψ⟩ (expressed as an inner product for functions, and then as a distribution/test-function dual pairing for general distributions). That identity motivates the extension to arbitrary distributions.

For a general distribution T ∈ D′ and a test function φ, the convolution φ * T is defined as a new distribution by specifying its action on test functions: (φ * T) is the distribution that sends a test function ψ to ⟨T, φ̌ * ψ⟩. The compact support of test functions ensures φ̌ * ψ is again a valid test function, so the dual pairing with T is well-defined. With this definition, convolution becomes a bilinear map: it takes a test function and a distribution to another distribution.

Finally, convolution on distributions inherits an identity element: the Dirac delta distribution δ behaves like a multiplicative “one,” acting as the neutral element for this convolution-based multiplication. That parallel to algebra is more than an analogy—it provides the structural foundation for later results, including how δ controls the behavior of convolution and why this framework is so useful for generalized functions.

Cornell Notes

Convolution starts as an operation on L1 functions: (f * g)(x) = ∫ f(x − y) g(y) dy, and it stays in L1 with the bound ||f * g||_1 ≤ ||f||_1 ||g||_1. To extend convolution to distributions, the definition shifts to how test functions interact with distributions via duality. For a locally integrable function f (a regular distribution) and test functions, swapping integrals leads to a reflected test function φ̌(z) = φ(−z). This yields the guiding identity ⟨φ * f, ψ⟩ = ⟨f, φ̌ * ψ⟩. For a general distribution T, φ * T is defined by (φ * T)(ψ) = T(φ̌ * ψ), producing another distribution. Under this convolution, the Dirac delta δ acts like the multiplicative identity.

How is convolution defined for integrable functions in L1(R^n), and what guarantee does it provide?

For f, g ∈ L1(R^n), convolution is defined by (f * g)(x) = ∫_{R^n} f(x − y) g(y) dy. The construction yields a new function on R^n that is again integrable. Moreover, it satisfies the norm estimate ||f * g||_1 ≤ ||f||_1 ||g||_1, showing convolution behaves like a controlled “multiplication” on the L1 space.

Why does the reflected test function φ̌(z) = φ(−z) appear when extending convolution to distributions?

When rewriting pairings involving convolution, the variables x and y get swapped after expanding the convolution inside an integral. Because the convolution uses the shifted argument x − y, changing the order of integration forces a sign change in the test function’s argument. The result is a reflected test function φ̌(z) = φ(−z), which is essential for the identity ⟨φ * f, ψ⟩ = ⟨f, φ̌ * ψ⟩.

How does the extension to distributions use duality rather than pointwise formulas?

Instead of defining (φ * T)(x) pointwise, the definition specifies how the new distribution acts on test functions. For T ∈ D′ and test functions φ, ψ, convolution is defined by (φ * T)(ψ) = T(φ̌ * ψ). This works because φ̌ * ψ is again a test function (compact support and smoothness are preserved), so the dual pairing with T is well-defined.

What role does compact support play in making the convolution with test functions well-defined?

Test functions have compact support, which ensures integrals involving products like φ(x − y) f(y) remain finite in the needed sense. That finiteness justifies exchanging the order of integration (a Fubini/Tonelli-type step) when rewriting pairings. Without compact support, the rearrangement could fail.

In what sense does convolution become an algebraic “multiplication” for distributions?

With the distribution-level definition (φ * T)(ψ) = T(φ̌ * ψ), the star operation takes a test function and a distribution to another distribution and is bilinear in the inputs. This mirrors multiplication in an algebra. A key structural fact is that the Dirac delta δ acts as the identity element for this convolution-based multiplication.

Review Questions

What is the exact L1 convolution formula and the associated norm inequality?
How does the identity involving the reflected test function φ̌ lead to the distribution-level definition of φ * T?
Why is it legitimate to swap the order of integration when rewriting the pairing for regular distributions?

Key Points

1
For f, g ∈ L1(R^n), convolution is defined by (f * g)(x) = ∫ f(x − y) g(y) dy and satisfies ||f * g||_1 ≤ ||f||_1 ||g||_1.
2
Convolution on L1 functions behaves like multiplication: it stays within L1 and supports algebraic rules such as distributivity.
3
To extend convolution to distributions, definitions shift from pointwise formulas to how operations act under dual pairings with test functions.
4
Swapping integration variables during the derivation introduces a reflected test function φ̌(z) = φ(−z).
5
For a general distribution T ∈ D′, the convolution φ * T is defined by (φ * T)(ψ) = T(φ̌ * ψ), producing a new distribution.
6
The Dirac delta distribution δ functions as the multiplicative identity for convolution in this distribution framework.

Highlights

Convolution for L1 functions is built from a mirrored shift: (f * g)(x) = ∫ f(x − y) g(y) dy, and it stays in L1 with ||f * g||_1 ≤ ||f||_1 ||g||_1.

Extending convolution to distributions relies on duality: (φ * T)(ψ) = T(φ̌ * ψ), so the operation is defined by its action on test functions.

The reflection φ̌(z) = φ(−z) is not cosmetic—it emerges from changing the order of integration when rewriting pairings.

Under the distribution definition, the Dirac delta δ plays the role of the identity element for convolution.

Topics

Mentioned

L1
D′