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Given a probability density function, $f\left(x\right)$, of a continuous random variable, $X$, and given an $N$-th order fourier series approximation:

$$f_N\left(x\right)=\sum_{n=-N}^{N}c_n e^{inx}$$

Is there an upper bound for the truncation error, for an arbitrary $f\left(x\right)$? (given only the assumptions $f\left(x\right) \ge 0$ and $\int_{-\infty}^{\infty}f\left(x\right)\text{d}x=1$)

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  • $\begingroup$ I THINK you can write down the $L_2$ error, use the triangle inequality on that integral, and manipulate the terms to give you a bound, but I don't recall the details. $\endgroup$
    – rajb245
    Apr 25, 2013 at 18:25
  • $\begingroup$ Or invoke parseval's theorem in that integral. An IEEE paper from 1972 seems to have the details, "Bounds on the truncation error of periodic signals" $\endgroup$
    – rajb245
    Apr 25, 2013 at 18:31
  • $\begingroup$ Did you mean $\int_{\mathbb{R}}$? Then we can choose $f$ to have support away from $[-\pi,\pi)$... Please explain more. $\endgroup$ Apr 25, 2013 at 19:00
  • $\begingroup$ @AD.: If it helps, I'm interested in a pdf that is periodic, so pretend its on the unit circle and $\int_{0}^{2\pi}f\left(x\right)\text{d}x=1$ $\endgroup$
    – okj
    Apr 25, 2013 at 20:24
  • $\begingroup$ @okj Okay, then note that $\int_0^{2\pi}f(x)\,dx=1$ means $c_0=2\pi$. Also, what kind of an estimate would you expect? That is, for what norm $\|\cdot\|_\sigma$ should we estimate $$\|f-f_N\|_\sigma ?$$ $\endgroup$ Apr 26, 2013 at 5:19

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