# What is meant by “real-valued argument” in this context of the convolution operation?

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.

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.