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.
2026-04-03 02:30:04.1775183404
Square Summable functions
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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.