Suppose I have some piecewise smooth periodic function $f$.
Why does $f$ need to be bounded in order to have a Fourier series representation? Couldn't we consider the interval that it's unbounded as a finite point of discontinuities?
2026-04-24 21:28:53.1777066133
Why does a function need to be bounded in order to have a Fourier series?
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It doesn't have to be bounded. In order to define the Fourier coefficients (using the Lebesgue integral), you do want $f$ to be in $L^1$, i.e. $\int_I |f(x)|\; dx < \infty$ where $I$ is the interval you're using for your Fourier series.