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I have a problem that comes up from time to time in signal processing applications.

Let $f(x)\geq0\, \forall x$ and $g(x)$ be real functions with finite range and support.

Let $I(f(x),g(x)) = \int_{-\infty}^{\infty} \text{d}x\, f(x+y)g(x)$. The variables $x$ and $y$ are just dummies on the same line so I think it is ok to abuse the notation a bit here.

My question is, what is $\frac{\partial}{\partial f(x)}I$?

From first principles (limit definition of a derivative and Riemann sum), the best I can determine is $\frac{\partial}{\partial f(x)}I = \int_{-\infty}^{\infty} \text{d}x\, w(x+y) g(x)$, where $w(x)$ is a window function with the same support as $f(x)$. This result jars with me, because from matrix calculus I know that $\frac{\partial}{\partial \textbf{b}} \textbf{A}\textbf{b} = \textbf{A}^T$, and since the integral is effectively a sum then by analogy I would expect that $\frac{\partial}{\partial f(x)}I = g(x+y)$.

The first principles approach is shaky because it involves the interchange of two limits, of which which I'm not sure how to prove the validity. Unfortunately I'm not very well versed in the Lebesgue integral, would that make the problem clearer?

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