Consider the space $L^1(\mathbb{R})$ or more generally $L^1(\mathbb{R}^n)$. I saw a lot of functions which actually not belong to these spaces, easy functions as $x$ or $\sin x$. The reason why the do not belong to $L^1(\mathbb{R})$ in this case is due to its behaviour at infinity. They do not vanish at infinity. So my question is, are there classes of functions (I think functions with compact support are one of them) which belongs to $L^1(\mathbb{R}^n)$? I would like to get a better feeling for which functions are in $L^1(\mathbb{R}^n)$ and which are not. It would be also nice, if one could give me a few examples of functions which belong to these spaces.
Functions belonging to $L^1(\mathbb{R}^n)$
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real-analysis
measure-theory
2 Answers
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As an exercise, I encourage you to look at \begin{align} f(x) = \frac{1}{(1+|x|)^\alpha} \end{align} and determine for what values of $\alpha$ will $f \in L^1(\mathbb{R}^n)$.
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As you correctly mentioned, the functions which are continuous and have compact support are contained in $L^1$. (IVP)
I want to introduce to you the Schwartz space $\mathcal{S}(\mathbb{R}^n)$. This space contains rapidly decreasing, smooth functions - take for example $e^{-|x|^2}$.
$$\mathcal{S}(\mathbb{R}^n):=\{f:\mathbb{R}^n \to \mathbb{C} ~:~ ||f||_{a,b}=\sup_{x \in \mathbb{R}^n} |x^b f^{(a)}(x)|<\infty \text{ for all } a,b, \in \mathbb{N}_0^n \}$$
This space is even dense in all $L^p$ spaces for $p\neq \infty$.
Let's just show $\mathcal{S} \subset L^1$:
$$\int_{\mathbb{R}^n} |f(x)| \ \text{d}x = \int_{\mathbb{R}^n} |f(x)| \frac{1+|x|^2}{1+|x|^2} \ \text{d}x \leq\left( ||f||_{0,0}+||f||_{0,2} \right) \int_{\mathbb{R}^n} \frac{\text{d}x}{1+|x|^2}<\infty $$