Kernel of the antiderivative map #

Background #

In class, we studied linear transformations on vector spaces of functions, finding that if \(K:\mathbb{R}\times\mathbb{R} \rightarrow \mathbb{R}\) then the map

\[ \mathcal{T}_K(f)(x) = \int_{-\infty}^\infty f(y) K(x,y) \; dy\]

which takes a function \(f\) to the function \(\mathcal{T}_K(f)\) defines such a linear transformation. Different choices of \(K\), called kernels, beget different transformations.

An example #

We found in class that the antiderivative from the Fundamental Theorem of Calculus is an example of a linear transformation. More specifically, given a continuous function \(f\), the Fundamental Theorem of Calculus defines another continuous function \(F\) via:

\[ F(x) = \int_{0}^x f(y) \; dy\]

We will write \(\mathcal{I}(f) = F\) for the map that takes a function to the above antiderivative.