4

Can somebody please help me understand the notion of square summable functions intuitively?? I have been self studying Hilbert Spaces and Fourier Transform for DSP. Any help is appreciated. Thanks.

Kalpan
  • 383

1 Answers1

7

A square summable function $f$ is one where $\int_{-\infty}^{\infty} |f(x)|^2 dx < \infty$.

Think about functions that violate this. Any function that goes to infinity (e.g. $f(x)=\frac{1}{x}$) is not square summable. But even nicer functions violate this. For example, any non-zero constant function (e.g. $f(x)=1$) is not square summable.

So square summable is a relatively strong condition. Not only do functions need to go to zero in both directions, but they have to go to zero "quickly" enough to be square summable.

I hope this help.

NicNic8
  • 7,120
  • 2
    The function $f$ defined by $f(x) = |x|^{-1/4}$ if $0 < |x| < 1$ and $f(x) = 0$ otherwise is unbounded near $0$ but is square summable. – littleO Jun 09 '17 at 06:08
  • "Square summable" is is just another way to say "square integrable", right? Strictly speaking, shouldn't the former only be applied to sequences rather than functions? https://en.wikipedia.org/wiki/Square-summable – Jess Riedel Sep 10 '20 at 18:34