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Let a function $f(x)$ be continuous, periodic, differentiable, and square-integrable on an open interval $(a,\,b)$. Will, its Fourier series coefficients change if it had all these qualities for a closed interval $[a,\,b]$. What about half-open and half-closed intervals $(a,\,b]$? What would be the difference in the series of the functions for all these intervals?

Muzammil
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    The answer is no and this follows from the fact that singleton sets have Lebesgue measure $0$. – Kavi Rama Murthy Oct 10 '21 at 04:38
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    Can you please elaborate on this a little? Or link an existing question for my clarity? – Muzammil Oct 10 '21 at 04:56
  • For any square integrable function f on $[-\pi, \pi]$ the Fourier coefficients are $\int_{-\pi}^{\pi} f(t)e^{int}dt$. For any other interval the coefficients can be written down by a similar formula. The integrals here do not depend on whether the end points are included or not. – Kavi Rama Murthy Oct 10 '21 at 05:21

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