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Consider the following statement from Deep Learning book (p. 327, chapter 9: Convolutional Networks)

In its most general form, convolution is an operation on two functions of a real-valued argument.

Suppose $f$ and $g$ are functions on which I want to apply convolution operation. What is meant by two functions of a "real-valued argument" in this context?

Does it mean $f$ and $g$ are real-valued functions? Or does it mean $f$ and $g$ are real functions? or any other?

  • Real-valued function: Function whose codomain is a subset of real numbers

  • Real function: Function whose domain and codomain are a subset of real numbers.

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In its most raw form, convolution is defined as: $(f*g)(t) = \int_{-\infty}^\infty f(\tau) \cdot g(t-\tau) d\tau$.

Here, t doesn't represent the time domain. Infact, it represents the real valued argument the book is talking about. In this notion, at moment t, convolution can be thought of as a weighted average of the function $f(\tau)$ weighted by $g(–\tau)$, which is simply shifted by amount t.

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  • $\begingroup$ Oh, the domains for both f and g are a subset of real numbers... t is continuous... $\endgroup$
    – hanugm
    Commented Apr 28, 2021 at 6:08

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