# Is it possible to represent the derivative operator as an integral transform?

Apparently, the Schwartz kernel theorem states that all linear operators can be represented as integral transforms (but only if you use generalized functions such as the dirac delta as kernels.) Representing the derivative operator as an integral transform would be really useful because then when defining strange and new generalizations of integrals one could automatically derive an appropriate generalization of derivatives to fit.

$$k(x,y) = -\delta^\prime(x-y)$$ where $\delta^\prime$ is the derivative of the delta distribution, defined in the distributional sense i.e.
$$\langle \delta^\prime(x-y),\varphi(y)\rangle = -\langle \delta(x-y),\varphi^\prime(y)\rangle = -\varphi^\prime(x)$$
• An additional information (that @Steven Stewart-Gallus has implicitely mentionned by citing a - disappointing - Wikipedia article) is that there is nice physical interpretation of $\delta'$ as modelizing a "doublet" (two opposite charges $q^+q^-$ that are infinitely close. Jun 6, 2016 at 21:55