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Does a function $f(x):[0,\infty)\rightarrow R$ with $f\in L^p$ for $p<\infty$ have to die down to $0$ as $x\rightarrow \infty$? I some how feel that the $L^p$ norm exist only when the function dies down to $0$ and the rate at which it dies down to $0$ depends on $p$. Is this right?

Math1000
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nadurthi
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    This is wrong the way you put it: standard counterexamples are "spiky" functions, which do not decay but belong to whatever $L^p(\mathbb{R})$ space you like (with $1\le p < \infty$). Nevertheless, that's a correct idea: somewhat more precisely, I would say that belonging to some $L^p$ space imposes an integral condition of decay at infinity. The bigger the exponent the stricter the condition. – Giuseppe Negro Dec 14 '13 at 01:19
  • "spiky" in the sense that f= 1 for integers and 0 every other real number. This function exists in Lp space but does not decay. Is this right? – nadurthi Dec 14 '13 at 01:21
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    for example a triangle at each integer of height $n$, width $1/n^3$ , doesnt "decay" in the sup norm, but the integral is small outside of large enough bounded sets. – yoyo Dec 14 '13 at 02:11
  • This is a great question, its a shame more people didn't answer. Thanks Giuseppe and yoyo. I also found this related question. – Diego Jul 10 '15 at 21:49

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If by “die down,” you mean “converge to $0$”, then the answer is no. Consider the following function: \begin{align*} f(x)\equiv \begin{cases} 0&\text{if $x\in[0,1)$,}\\ 1&\text{if $x\in[1,1+1/1^2)$,}\\ 1&\text{if $x\in[2,2+1/2^2)$,}\\ 0&\text{if $x\in[2+1/2^2,3)$,}\\ 1&\text{if $x\in[3,3+1/3^2)$,}\\ 0&\text{if $x\in[3+1/3^2,4)$,}\\ \vdots\\ 1&\text{if $x\in[n,n+1/n^2)$,}\\ 0&\text{if $x\in[n+1/n^2,n+1)$,}\\ \vdots \end{cases} \end{align*} This function doesn't converge to zero as $x\to\infty$ (nor is it equal almost everywhere to a function that does), yet, for any $p\in[1,\infty)$: \begin{align*} \|f\|_p=\left(\int|f(x)|^p\,\mathrm dx\right)^{1/p}=\left(\sum_{n=1}^{\infty}\frac{1}{n^2}\right)^{1/p}=\left(\frac{\pi^2}{6}\right)^{1/p}<\infty. \end{align*}

triple_sec
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