Colombeau algebra

Such a multiplication of distributions has long been believed to be impossible because of L. Schwartz' impossibility result, which basically states that there cannot be a differential algebra containing the space of distributions and preserving the product of continuous functions. However, if one only wants to preserve the product of smooth functions instead such a construction becomes possible, as demonstrated first by Colombeau.

As a mathematical tool, Colombeau algebras can be said to combine a treatment of singularities, differentiation and nonlinear operations in one framework, lifting the limitations of distribution theory. These algebras have found numerous applications in the fields of partial differential equations, geophysics, microlocal analysis and general relativity so far .

Colombeau algebras are named after French mathematician Jean François Colombeau.

Schwartz' impossibility result

Attempting to embed the space ${\displaystyle {\mathcal{D}}^{\prime }(\mathbb{R})}$ of distributions on ${\displaystyle \mathbb{R}}$ into an associative algebra ${\displaystyle (A(\mathbb{R}),\circ ,+)}$, the following requirements seem to be natural:

1. ${\displaystyle {\mathcal{D}}^{\prime }(\mathbb{R})}$ is linearly embedded into ${\displaystyle A(\mathbb{R})}$ such that the constant function ${\displaystyle 1}$ becomes the unity in ${\displaystyle A(\mathbb{R})}$,
2. There is a partial derivative operator ${\displaystyle \mathrm{\partial }}$ on ${\displaystyle A(\mathbb{R})}$ which is linear and satisfies the Leibniz rule,
3. the restriction of ${\displaystyle \mathrm{\partial }}$ to ${\displaystyle {\mathcal{D}}^{\prime }(\mathbb{R})}$ coincides with the usual partial derivative,
4. the restriction of ${\displaystyle \circ }$ to ${\displaystyle C(\mathbb{R})\times C(\mathbb{R})}$ coincides with the pointwise product.

However, L. Schwartz' result implies that these requirements cannot hold simultaneously. The same is true even if, in 4., one replaces ${\displaystyle C(\mathbb{R})}$ by ${\displaystyle C^k(\mathbb{R})}$, the space of ${\displaystyle k}$ times continuously differentiable functions. While this result has often been interpreted as saying that a general multiplication of distributions is not possible, in fact it only states that one cannot unrestrictedly combine differentiation, multiplication of continuous functions and the presence of singular objects like the Dirac delta.

Colombeau algebras are constructed to satisfy conditions 1.–3. and a condition like 4., but with ${\displaystyle C(\mathbb{R})\times C(\mathbb{R})}$ replaced by ${\displaystyle C^{\mathrm{\infty }}(\mathbb{R})\times C^{\mathrm{\infty }}(\mathbb{R})}$, i.e., they preserve the product of smooth (infinitely differentiable) functions only.