Anyone got any idea how I could find a closed form expression for this sum:

$$f(x) = \sum_{\nu = - N}^{N} \big(N - |\nu|\big) \lambda^{|\nu|}e^{i 2 \pi \nu x}$$ where $\lambda \in [0,1]$?

I notice that it looks similar to an expression for the Fejér kernel.


$$S=\sum_{k=1}^n kz^k=\sum_{k=0}^{n-1} (k+1)z^{k+1}=z\sum_{k=0}^{n-1} kz^k+z\sum_{k=0}^{n-1} z^k=z(S-nz^n)+\frac{z^k-1}{z-1}.$$

This should be enough for you to derive the complete formula with

$$f(x)=N+N\sum_{\nu=1}^N(\lambda e^{i2\pi x})^\nu-\sum_{\nu=1}^N\nu(\lambda e^{i2\pi x})^\nu +N\sum_{\nu=1}^N\left(\frac1{\lambda e^{i2\pi x}}\right)^\nu+\sum_{\nu=1}^N\nu\left(\frac1{\lambda e^{i2\pi x}}\right)^\nu.$$

  • $\begingroup$ Very nice, thanks! $\endgroup$ – Allen Hart Sep 21 '16 at 8:06

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